Disclaimer: I am not a vet, this is not medical advice, seek professional guidance before taking advice from the internet.

Technically, kind of. It’s extremely rare, but cats can develop Fel D1 sensitivity, resulting respiratory issues. Usually not to their own antigens, but another cat’s. I think technically that would be considered an autoimmune condition if they were allergic to themselves. Probably would present as something similar to pemphigus foliaceus.

Yes. We got our dumb autoimmune problems, they got their own.
The problem is mostly environmental, all kinds of toxic shit which screw us up inside and out.

I don’t know, but I’ll note that, if you are asking after seeing a cat fussing over an irritating area, you might be confusing cause and effect. That is, irritation due to some other cause could be driving the cat to lick/groom that area.