1. Why did Inara leave Sihon? (sp?)
Somewhere, I got the impression that the Companions' ...Academy(?) or whatever it was called was primarily a training facility, and once a student graduated, she, or he, if male Companions existed, were pretty much expected to leave and ply their chosen trade somewhere else, either operating from a traveling base, like Inara did from "Serenity," or at a permanent base on another planet or moon.
2. What did Wash, Kaylee and Jayne do during the war?
I got the impression that the Rebels weren't a huge force, so probably the greatest majority of the Federation's population weren't involved in the fighting at all.
3. How did Wash and Zoe get together?
Wash was hired by Mal to be the ship's pilot, so he and Zoe probably didn't know each other before that, and it was a case of dislike at first site, for Zoe, at least, as was shown in "Out Of Gas." My guess is that she didn't like him at first because of that ridiculous growth of hair on his upper lip, but once he shaved it off, it was a case of love at second site.