Two different topics in play here!
First, exposure. f numbers are expressed as f/2 etc for a reason - they are fractions. The diameter of the lens opening to the focal length in fact. f/2 on a 100mm lens has an opening diameter of 50mm; f/2 on a 50mm lens has an opening 25mm wide. Why? Because of two laws of physics.
The amount of light passing through a hole - like the amount of water through a pipe - becomes greater as the diameter increases. The increase depends on the area though, and the area is proportional to the square of the diameter. The second law is the inverse square law. Double the distance a torch beam travels (torch because you can perform the experiment at home) and the diameter of the beam on the wall doubles. Which means the area quadruples; and as extra light isn't attracted by gravity into the beam of the torch as it crosses the room (nobody split hair here about gravitational effects!) this means that the brightness has actually not halved but quartered. That's the inverse square law.
The second part of this is that to focus on infinity, a 100mm lens has to be 100mm away from the film plane. A 50mm lens only 50mm. So the amount of light passed for a given lens opening has quartered at each image point. So we have to let 4 times as much light in by doubling the opening. Which is why f numbers are defined in terms of fractions (or ratios if you're continental).
Depth of field is a separate topic. A perfect lens reproduces a point in the object as a point in the image. Unfortunately, there is no such things as a perfect lens. And even if there was, physics has another trick up its sleeve to prevent imaging perfection. Diffraction. Light is a wave motion, and as a result the image produced by a perfect lens isn't a point but a disc. Look up Airey disk. So we never have a point, always a (small) circle. If the eye accepts it as a point. all well and good - it looks sharp. With a perfect lens, the image rays from the focus point are brought to a point at the film. The image from closer focuses behind the film (or would, if it got through) and so hits the film while still converging, giving a circle not a point. Light from further away than the focused plane meets in front of the film, so is diverging again when it hits the film. Another circle, not point.
As it's all down to what the eye will accept, depth of field will depend on things outside the pure physics of this explanation. First, how good is your eyesight? Second, how closely are you examining the image from? And how much have you enlarged the image? All of these subjective factors are combined with the maths to produce depth of field tables.
The interesting point here is that for the same size of print a 35mm negative is enlarged more than a medium format one. A longer focal length lens will have shallower depth of field, so a medium format negative taken with a standard lens (let's say from a distance that with that lens shows a half length portrait) would exhibit shallower depth of field than a 35mm image with a standard lens taken from the same position. But the 35mm image would contain more extraneous matter, and to arrive at an identical image size in the print would need enlarging to not only use only the central part, but also to match the print size from the larger negative. What effect does this have?
Someone asked this as a question to Victor Blackman in the 1960s, and he devoted a column to answering it. The punchline was a pair of photos, one from medium format and one from 35mm taken from the same position of the same seated person. Enlarged to show the upper body at the same size, the depth of field appeared to be actually the same - the extra magnification just about canceled out the shallower depth of field.
I haven't checked the equations, and I haven't repeated the experiment, but it is worth considering...
