Product manager: “Let’s say, the development team comes back to you and says ‘Hey this looks great but we can’t do this’. As a UX designer how would you handle this?”
I'd demand that management realize that this is the primary problem with waterfall design methods and that it's imperative that we bring development into the design process much earlier than is happening.
Product manager: “Developers are sensitive to making changes to a feature because they have so much vested time involved with a project. How would you go about making change in this political environment.”
See above. This is a management problem. Dev needs to be on the same team as UX. We're not in competition, but partners. Any time that dev invests should be the same time UX is investing so that everything is a joint endeavor rather than an us vs. them.
HR manager: “What if I came to you and said ‘I don’t like the way that button looks or how that interaction functions’?”
I'd say "who cares what HR thinks? What's our user research show?"
Ok, as you can probably figure out, those aren't the exact answers I'd use. Those are the answers I'd like to reply with, but realize that probably wouldn't go over well.
Bottom line, if I were asked those specific questions in an interview, I'd walk away from that position. It's clear to me that half the time would be spent with political battles. No one's got time for that.