#1
Just as title states will changing the action height at the bridge throw out the intonation a bit?
#3
Theoretically, yes of course.

Practically, yes of course.



...I mean, yes, it will. But, how much it'll change depend on how much you lower or raise the action. Just set your action to where you want to have it, then worry about intonation. As the person above me said, just check it and if it's too far out of whack for your comfort, adjust as needed.
#4
I lowered my action on the high e string a bit, it needed a slight intonation adjustment but now its is dialed in perfectly. Still messing around with action height to see where I like it, feel the most comfortable.