I think you should. You'd be surprised at how your calm and respectful approach can eventually change someone's mannerisms, ideas and attitude. Why fight fire with fire?

Yes. I believe you should (almost) always be calm and civil when dealing with someone in authority or anyone really. In my experience, being civil always helps fix a situation more than being angry and disrespectful. But I won't just bow down and do as I'm told if i don't agree. But I'll respectfully explain why I don't agree.

No. I'm not going to be respectful to someone who doesn't respect me in the first place. It's a two way street, if you start an interaction being rude and hateful, you can expect that feeling to be returned.

Just be an adult. Not everyone likes everyone, but you can be civil. Someone can be super disrespectful to me, doesn't mean I need to react.

If you don't show respect you don't get respect from me.

If you're being disrespectful to me I'm going to be disrespectful to you.

I try being respectful in all situations, even if I don't agree. I will speak up, I'm not a doormat, but I try finding the best way to present my argument. I think people who are rude/disrespectful from the very beginning of an interaction just because they disagree with you are childish.