Do you mean a lens with an actual focal length of 250mm or a lens with a focal length of 90mm (250mm equivalent for a 2.7X crop factor)? There's a big difference between the two.
But a 90/2 will behave like a 250/5.4 on a FX camera, which would be a longish, thinnish lens. To work like a 250/2 it would have to be a 90/0.75, which is not impossible. the closest I can think of is the famous Zeiss Planar 50/0.7 used by Stanly Kubrick for Barry Lyndon
Only for purposes of approximate depth of field, not for exposure. Nor would it be any longer than a 90mm f/2 for an FX camera, although some of the rear elements could be made lightly smaller.
The minimum size of the front element is an absolute number. It can't be smaller than the focal length divided by the f/number. That's how the light gets in and you need the area. So a 250mm f/2 has to have a front element at least 125mm in diameter regardless of the field.
As I noted, the front element has to be at least as big as the focal length over the f/number. But a lot of SLR lenses are retrofocus. I have a 10-24 Nikkor. This lens is f/4.5 at 24mm, which suggests that the front element has to be 5.3mm across. But it's actually 47mm across so that the lens can cover 109 degrees without hitting the reflex mirror. So in this case, the front element has to be big enough to support the minimum focal length.
A 250mm f/2 must have a front element at least 125mm in diameter. Otherwise it wouldn't be an f/2. A 250mm lens has no trouble covering FX or, for that matter, 6X6. So you don't save anything by limiting the image circle. Note that there are very few DX lenses above 35mm; you just don't save a whole lot. At short focal lengths, it is easier to make a DX lens than an FX lens but you still have the same mirror clearance, making DX wide angles fairly large themselves. The Nikon 1 system eliminates that, so you could have really tiny wide angles. And a system with a 2.7 crop factor allows you to use shorter lenses to get equal fields of view.
Who cares about exposure? It is the same as the f/5.4 for the amount of light forming the image, which is what sets the noise, so the slowest shutter speed that you can use. In any case, at normal subject distances the DOF is the same to a good degree of precision.
The amount of light forming the image is exposure, so it sounds like you care about it. An f/2.0 lens is an f/2.0 lens, no matter how big or small an image circle it covers.
That's where you are wrong. The amount of light forming the image is the exposure times the area of the sensor. Exposure is measured in lux-seconds, and lux is lumens per square metre. To get amount of light (lumens) you need to take the area into account.
Certainly it is, but the amount of light that goes into the image depends on the f-number and the sensor area, not just the f-number.
