(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: