Tree
No, binary or general search trees questions most of the time. You may have to combine trees and heap/stacks for some interviews though.
No.
For backend roles yes. Never for front end
No, in my experience with over a dozen interviews and on-sites at Big N and unicorns, I've never been asked it once. Anecdotally, array and hash tables, DP, and graph/topological sort seem to be "in" these days.
+1 on this. Especially graph and topological sort. I've been given lots of traverse some matrix problems in interviews at different places
Yes, I faced one time and unexpectedly i was able to write a full functional solution using RB tree..
Yes. STL data structures use these so you're expected to know this if you're interviewing for a dev position. Tree balancing is very basic CS knowledge.
O_o
This is not basic cs knowledge! Almost every textbook defers this topic to advanced chapters or appendix. It's also fairly useless knowledge imo