English doesn't have formal and informal speech the way other languages do. When we say a sentence is formal, we usually mean it's too awkward, old-fashioned or academic (as opposed to being something said in formal situations). "Perplexed" isn't formal because it's reserved for formal situations. People say "I was perplexed" sounds formal because it sounds more awkward/academic than "I was confused", just because "confused" is far more common than "perplexed". It's not really a big deal, but what they found funny might have been that an English learner was using an uncommon word. I think it's very unlikely that it had anything to do with being in or out of the classroom. If an English speaker says you're speaking formally, they probably don't mean that your speaking in a way reserved for formal situations, but rather that you're speaking in an awkward or roundabout way.

It depends on who you're speaking to. Most of the time we only speak formally for business meetings/interviews etc. You should be able to tell from the situation which is more appropriate to use. Just listen to how others are speaking at the time.

I don't know why they laughed. Overuse of formal language sounds tacky. Don't overdo it!