(1) Yes, I think that's one of the ways to define schemes. Look for `representable functors` and you'll get lots of literature. 

It was a crazy idea about 50 years go, part of establishment nowadays.

I'm not an expert, but I think in (3) it's crucial that rings can be localized. I think there's some notion of localizability in category theory and it boils down to something *any localizable thing is a (subthing) of sheaves on a site* (the formal statement is "any presentable category can be obtained as a localization of some category of sheaves of sets").

For (4) I think the situation is quite simple. Schemes are easy to imagine for most people, so people work in scheme language unless there's a need for more general topoi.

Here are also my earlier questions:
 
* [What is a topos?][1]
* [How to think about model categories?][2]


  [1]: https://mathoverflow.net/questions/101/what-is-a-topos
  [2]: https://mathoverflow.net/questions/2185/how-to-think-about-model-categories