Not at all, just look at the difference in the current sensors on the market. There's a wide range of performances due to engineering restrictions. Only the pixel size and aperture size are really relevant to diffraction so it occupies a different class of problems.

Even if you had 100% efficient sensors, photon shot noise would still be an issue. You are sampling the photons emitted by the target. After having reasonably efficient sensors, it's a baseline you can't go under.

I'm not sure about that minimum, but I do know noise (dark current signal non-uniformity for those in the know) increases considerably with temperature. So yes it matters.

FWIW, the rods in your own eye's retina are able to register single photon impacts.

Ok this is getting pretty convoluted. I didn't mean to imply that photon noise was not a real thing, just that diffraction is a more direct concern. Photon noise can be amortised by a number of different processes which aren't applicable to the case I was trying illustrate. You can't amortise diffraction errors by longer exposure or better photon capture. Pixel size is a primary concern regardless of its noise characteristics.