Oh-Kay... I am going to propose a completely alternatve 'solution'..... I really dont think you need another lens, here. What I think you need is to get out the house.
This is not cheauvanism, I was a house-husband, and then a single parent. Kids aint easy... juggling kids, housework and a camera is a mamoth task.... BUT, seperate the variables, do one thing well, rather than half a dozen badly, if at all!
Take childlings to the park, take photo's there.. or some-where.. ball-pit, wacky-ware-house, the woods, or whatever.
The issue of framing with the 50 is significantly that in the confined space of a house you just dont have the room to back up enough to get the all in frame.. so take them some-where you do. Take them out-doors, you hopefully should also have a heck of a lot of extra light, so the matter of the fast aperture and debate over shutter speeds and ISO levels, should become less pertinant, too.
BUT... you get out the house. You dont have laundry to do whilst you peel the potatoes, and a microwave yelling you the chops have defrosted, and dish-washer incessantly bleeping for attension and its salt dish filling, while childling one is screaming childling two has turned over pokemarble-teenage-pregnant-pigs, (or whatever the must watch kids show is right now!), and you can hear the destruction of preciouse artifiacts occurig (if they are still young enough there are any left!!!) or childling three, is going through the kitchen cupbards insisting teacher said they had to bring a packet of raisins to school tomorrow.. etc etc etc... and unabated chaos to deal with similtenousely.. while you really want to get a photo!
Go OUT!
That all goes away. Kids let off steam, so they are more likely to eat thier cabbage than chuck it at the walls, when they get back, and they do stuff they wouldn't in the house, that is usually more photogenic, A-N-D you have all the space you need, and scenary you dont have to hoover when you have done!
When you get them home.. it will almost inevitably NOT be as late as you had feared, but, having had some let-of-steam time, kids will almost certainly settle more readily and do home-work, or whatever, in a far more relxed and responsive frame of mind, and leave you alone to get on and do those jobs you have to, like cook dinner!
You may, in an incredibly unlikely fluke, feeling like they have had a treat, utterly unprompted, let alone nagged for half a week, say "I brought the washing down from the bedroom Mum*.. all of it! Yes every-ones! Shall I put it in the washer for you?"
My advice is say "Yes", but dont let them add soap or put the machine 'on'! You can take it all out and sort reds from lights, and soap and settings later, like when they are in bed; just be greatful not to have to hunt! (Though there will inevitably be a pair of underpants hidden in a crevice some-where, they missed!)
* I know... I'm a bloke... AND I 'think' kids did... b-u-t...easy mistake to make, especially by kids.. but not as bad as when they start to say one and finsh saying the other, when my name would be 'Mad' which is only slightly better than 'Mud'... which I am probably more used to TBH! Lol! Bit of advice shared by generals and parents.... pick only the battles you need, and can possibly win! The rest? Let it slide!
Point is, that a new lens wont make your house any bigger; nor will it make it any better lit. Nor will it make your kids any more co-operative.
Change of venue can.
And it wont just have possitive effect on your photo's!
B-U-T this is where you start taking 'control' of photography, not looking for f-stops, or focus schemes, or ISO levels.... Plain and simple setting the scene.
If you had warehouse sized studio to shoot in, same deal would be at the start, long before you looked at the camera or what lens to strap to it. Setting up the scene you are going to photograph your subject in; what furnature, what back-ground, what props, and the be-all and end all, what LIGHT.
First, last and half the intermediete questions that one. Indoors, outdoors, in my ladies chamber, LIGHT is all.
Outdoors, its the sun! Big, bright and free! During the day. Often not all that co-operative, and its a bit hard to move around... but not uncontroleable! You can use shades to reduce it in some places, reflectors to boost it in others; and this needn't be complicated and expensve either.
On the spot, improvisation, I have used my trench-coat to shade one of my kids from direct sun in a photo, or I have placed a white T-shirt or light beach towel over a car door, or window or hung from a tree as a reflector, to put light where I want and not where I dont, to help light my subject as I preffer.
This has been on beach, or at the country park, or wherever, out-and-about, not lugging a bag of e-bay studio accessories around with me on the off-chance along with the havasack of drinks and crisps and elasto-plast, spare jumpers and wot-not necessary for a day out with the darlings!
Its emnantly feasible to take control of lighting your scene like that, with a little imagination and improvsation, and just a little know-how.
If you are working in your own home; even more oportunty to control the light, whether turning on and off ceiling lamps, or hunting out a bedside light, or desk light, to chuck a bit more where you want it, or simply opening or closing a curtain!
This is all very very do-able, wherever you set up; it doesn't take any new gear or a shopping trip, and likely very little faffing about...
BUT this is where you really start to 'go manual' and genuinely take control of your photos, rather than play cameras, prodding buttons... and a log long time before even looking at the camera, let alone, choosng what lens, less still what 'settings'!
And, when it comes to artifcial lghting, D3200 has an inbuilt flash, you alrady have ONE, IF you choose to use it.
Personally I am not a big fan of flash, and the uflattering direct lighting of on-camera flash that's so often critised, but it is there, you do have it, you may use it.
And again, a little imagination, know how and improvisation can go a long way. I have used hankerchiefs or tissue paper as flash diffusers before now; I have folded a piece of paper as a 'snoot'. I have bounced flash off the cieling or off a sheet... its light, and just like the sun, you can effect control where it goes, with improvisation and available materials as easily as with a big box of studio equipment.
All this sort of thing is possible with a little know-how, a little imagnation and a bit of improvsation. NOT a fancy camera, or amazing lens or other wonder-widget.
Heck, taking pictures of my own kids, an alwful lot of the time I used this sort of stuff, to get 'better' photo's with nothing more than a cheap compact, usually fixed and fixed focal length lens, on either film or widgetal!
I can feel a lot of stammered excuses and half rasons why you think, that you 'have' to find the perfect les to do what you think it should, the way you would preffer, backing up, wanting to be chucked in here... BUT no buts!
So much is in your aproach and expectation.. and the point is its better photographers that make better photo's not better cameras or better lenses, or greater gadgets.
I have the exact same camera, a Nikon D3200 as you, and I am certain that you wont actually solve many if any of the problems you suggest your suffering with ANY other lens.
I have used the Nikon Auto-Focus 'prime' 50, I bought my daughter on that camera. I can well apreciate the brighter view-finder it offers, from its f1.8 max aperture.
But for low-light photography, meh! The extra couple of stops it offers is no real big deal, Its the difference between setting ISO3200 rather than Hi1 ISO boost, or being able to use a shutter of 1/30th instead of 1/8th...
And if you have to go that high on ISO and or that low on the shutter, you are hard up on the buffers, before you begin, and the issue isn't lack of f-numbers but fundementally, simply the lack of light!
High ISO noise, if it's an issue isn't going to be combated much if at all, by the faster lens, that high up the ISO amp-ramp.
Motion blur, whether from camera shake, or subject movement, similarly, isn't going to be resolved, by being able to use what would still be a petty slow shutter, especially with quick-kids darting about the frame!
The Kit 18-55, also has image stablsation as some aid for hand-holding, that the prime 50 doesn't; not that I believe it it actually helps any if you have decent hand-holding technique to start with.
SO!, main advantage of that lens, if there is one, is shallow focus effects. Which, is a pet gripe of mine about fast prime auto-focus lenses. So often used to get razor thin DoF, very shallow focus. When a bit of know-how could be applied, and more strategically, using much more moderate apertures, and a bit of manual focus used to put the DoF where you really want it in a scene to get an in focus subject and out-of-focus back-ground and probably avoid the problem of getting blury noses and ears, puting the red-dot on the eye, and leavg the rest to the electrickery! So even THAT one-trick dog is no huge or real 'advantage' very often.
Big dissadvantage is that on a crop-sensor body, it does give a narrow angle of view. Its focal length fixed, and at the top, telephoto end of that zoom's range.
This will mean that you have to back up a fair bit to get a subject in the frame and not chop half thier head off, compared to a wider angle, shorter focal length lens.
Closest focus distance, for the Nikon AF-S 50 is qoted as just 45cm or 18".. basically arms length. I really dont think that should be a very big problem.. I suspect the main one is simply that the angle of view means you have to back away so much further than that to get all your subject in the shot.
For note the Kit 18-55's closest focus distance is 30cm/ a foot, you could probably focus from closer range with that lens than you can with the prime! But you'd still be over flowing the frame at longer zoom settings!
The logical suggeston based on your apreciation of the 50, is the wider angle f1.8 AF-S 35 prime, I think has been mentioned. This has a closest focus disatance of just under 30cm/ 1foot, it doesn't actally focus much closer than the kit lens. But does frame wider.
Essentally, the only advantage of this lens, or any other wide angle, is merely the wider framing it offers.
BUT the trade off, is that reducing the focus range, you will, inherently reduce whatever DoF is avalable at selected aperture setting, working that close up, and with a wide aperture, you will run into the issue of focusing on eyes ad having noses and ears drop out of focus, but probably more significantly the perspective shift you can get, where near objects are rendered larger in the frame than far ones.
This gets more pronounced as focus distance gets shorter, and the ratio of the front-to-back distance between objects in the scene, becomes much larger in relaton to the subect to camera distance. Now with the wider angle lens, noses risk looking huge and out of proportion to the face, as well as out of focus!
If you must buy another lens, then the Nikon AF-S 35 is a cracking bit of kit, and very very good value for money.. BUT...
I would suggest that the wider framing it offers, will likely give you the 'idea' that its 'solved' all the problems you grumbled about, BUT, I also think that failing to get the best you could from the Kit 18-55 you already have, in reality, it would do little or nothing useful for you.other than entrench ideas of looking to the camera shop for an off the shelf solution to any problem you encounter, rather than craft.
As the 50, you would likely apreciate shallow focus effects it offers, but, using the lens in Auto-Focus, and achieving them purely from wacking the aperture wide open, its a one trick dog, you wont get the most from, and will as the fuzzy nose and ears example, make as many problems as it seems to solve.
Wider angle lenses still? Well, I have the estate agent's favourite, the Sigma 8-16 ultra-wide-angle, with its pretty fast, f2.8 max aperture.
This lens, s incredible for opening up and packing in everything in a small space. It is also pretty fast on aperture for a UWA lens.
But again, you are still up against copetig forces. With such a short focal length, the minimum focus distance is very close, but soo too is the hyperfocal distance, even with a wide aperture the scope for shallow or selective focus effects with a lens this short is pretty negligible.
It is a lens, unlikely to solve ay of the problems you suggest, other than backing into walls, and a heck of a lot of money, to make more problems than you had to start with. UWA's are icredibly tricky to make work well, with the perspective so dramatically effected by the short subject ranges, and very small changes of camera angle having huge effect.
I have to say SAVE YOUR MONEY.
My reccomend would be to put the 18-55 back on the front.
It is my most used lens on my own D3200 despite the alternatves available to me.
With that you have a heck of a lot more wide, if you need it, to avoid subjects overflowing the edges of the frame.
If the light levels are so low, that you are hard up against the ISO stops at ISO6400 or Hi1 boost settings, at f5.6, and you cant set a shutter speed higher than 1/15th... you are sat in the dark! I just turned the lights off and tried!
So your issue is simply lack of light, not lack of settings! You NEED flash, you NEED articical light! A different lens wont find that for you!
Change your aproach, not your lens!
With the 18-55, you have plenty of scope to get wider framing. The slower max apaetures are NOT a significant impediment in low light... the LOW LIGHT is the impediment! Tackle that.
Use craft. Open a curtain; use a table lamp! Use the ruddy flash thats on the camera!
On a wider angle lens the shallow focus you can get from the 50, wont be available, even if the same fast aperture is, as on the AFS35.
For 'situational' photography.... rather than calling it snap-shots (though there's a lecture there, I so oft say venerate, dont denegrate, the snap-shot!).. going wider, includng more in the frame, may reduce the impact of a frame filling face, but the inclusion of context, setting, etc, usually makes up far more, giving the picture meaning and relevence, and simple 'interest', here less zoom is so often more photo, and more detail begs more of it in focus.. if its oofed, it has as little meaning as if it wasn't there to start with.
Which is back to setting the scene. IF there is something in your scene you dont want in your picture... first course should be to see if you can simply move it! Not try and exclude it with tighter framing!
If you cant move 'it', then what about moving you? Change the camera angle! Again, costs nothing, and care and attension to whats outside the camera will make more dfference than to whats on the camera and looking for buttons to prod!
Back to aproach, and expectation, and using craft not kit to get what you want.
BUT... the biggest 'thing' is to change your aspirations and expectations. You want studio quality photos, get a studio to take them, or go to a studio, but expect them to be as perfectly boring as they are perfectly executed.
This sort of photography, call it situational, call it candis, all it sap shot, 'un-posed'... you are trading the formal and the contrived of a studio set up, for natural and sponteniouse, but you HAVE to expect the trade off, you cant have the natural and sponteniouse without the 'formal' set-up... you can get a lot.. with some craft and know how, tidyng the scene, placing lights or openng curtains, making the ost you can of that before shootng, but only so much, and if you do manage to do a really good job, the pendulum will often swing back, and you will loose a lot of natural and spontaniouse, for that apparent 'staged' level of excelence.
You can almost certaily get better pictures one way or another, we ALL could!... Question is how?
And here, I really, really dont think that you will get them from ANYTHING you can buy in the camera shop. I really dont.
You have more than enough gear already. That 18-55, is it. You use the know how to get better results from it. THAT is all... and if you struggle to follow on-lne tutorials.. meh.. yeah, they possible aren't the best tutorial, probably are jargon laden, BUT... answer is still in the know-how. Differet tutorials. Maybe a book. Maybe read the manual that came with the camera, or use the 'in camera' handy hints function!... its a bit terse and would you like fries with that, for my liking, but it IS there, and it is aother way to get know-how. Enrolling in a photo class may be another. Joning a photo-club aother still.
There are, as they say, many ways to skin a cat, you just need to find the one that best suits you. BUT, the here and now is to make sure you have hold of the right cat! Not the neighbors dog, or anything!!! And I think looking for another lens, is like trying to grab a chicken, in that analogy!