Well, I'd agree with May (or April) as a best bet. I lived on Skye for a couple of years recently, and still intend to return to build a house, once some pressing family matters are sorted.
Midges? Loads - but it depends upon your attitude and experience. Expect to be bitten unless its winter. Plan to minimise this by having the least amount of flesh available (tuck trousers into long socks!), avoiding damp airless areas (if you can see clouds of them, they'll annihilate you!). Wind is good - it blows them away. Sun is good - most biting species avoid direct sunlight. Expect a couple of hours after dawn, and the late afternoon/evening to be the worst for midges. If there are a lot, plaster your face & exposed skin with some cheap skin cream - nothing expensive and the scent is immaterial. The midges will land on this greasy surface ,stick to it and die. With a nice (??) thick greasy layer you'll gradually turn black as the midge corpses pile up, but very few will penetrate it to bite you!
Weather? Well, this really is pot luck. It is not unusual for a whole week to be wet and overcast on Skye. Weather forecasts are much less reliable than in most other locations - The Island makes it's own weather, especially if you have an air mass coming from the SW or W, over the sea.
But, in my experience, it is always worth bearing in mind that the weather on Skye can be dramatically different from one location to another. When we stayed (or lived, in English parlance!!) in Glen Eynort (between Glen Brittle and Glen Talisker) we'd often experience a very overcast, dour & drizzly day. All day. But if we drove 4 miles to Carbost, on higher ground, you could sometimes see a different story. Perhaps to the north-west it looked a bit brighter - in which case a drive up to Trotternish might be worthwhile (a long way, though, this is not a small island - so we really only tried this if we had visitors!). Otherwise a brighter aspect to the east might encourage a drive to Broadford and then down the road towards Elgol. Do not assume that crap weather over the Cuillin - or even very poor weather near the Cuillin, means that the whole Island is similarly afflicted!
Another quick observation. On Donnie's excellent panorama of the Cuillins you'll see a brighter area on the horizon just right of middle. Believe it or not, this is almost always there! This gap is Glen Sligachan, running pretty much between the Red Cuillin and the Black Cuillin. I've lost count of the times that the Black Cuillin are shrouded in cloud, the Red Cuillin may also be, or swirling mist, but there is a very bright and interesting area of sky between the two. It can be very interesting at sunrise, but the area is so changeable that a striking image is possible at any time of the day.
Hope this helps.
To go back to time of year, yep May would be my pick. Otherwise, September (watch the midges!), or October if you could stretch it.