Yes. Sometimes it's good to use 'pozostać' to make it clear that you mean 'stay'.
"zostać" can technically mean "to remain", but rather only when it's the only interpretation that makes sense. Otherwise, use "pozostać", it's definitely "to remain".
Means both, depending on the context. In fact, I'd say that "became" may be the main meaning, as when you want things to be unambiguous and definitely mean "remained" or "stayed", that's "pozostał".