Yes, 물론이죠/물론이지 is slightly formal. I don't think I've ever used 그럼요 or 당연하죠 with teachers or professors; they both are informal. 당연하죠 is closer to "You bet!". But as for 물론이죠, I think I've used it with them. 당연히 is an adverb. I think you can take it as "Of course, ..." in some cases, "naturally/absolutely" in other cases. A : I need a book called HGG, do you have it? B : 당연히 있지. Of course, I have it. A : She is mad at me, and I don't know why. B : What did you say to her? A : I said she got fat. B : 여자한테 살쪘다 그러면 당연히 화내지! If you tell a woman that she got fat, she will absolutely get mad! 당연한 is an adjective. A : I'll never tell you a lie. B : 당연히 그래야지. 당연한 말을 왜 해? Of course, you must not. Why are you saying the obvious? I do say some phrases other than the three, but they are not really useful. 당근이지 is a slangy version(just because 당근이지 and 당연하지 sounds similar; 당근 is a carrot). Other ones are dialectic expressions.

It think it depends on its suffix. 그럼요, 당연하지요(당연하죠), 물론이지요(물론이죠) are considered polite/formal. 그럼, 당연하지, 물론(이지) are casual. I think I use 그럼 and 당연하지 more often simply because 물론's harder to pronounce.

그럼요 is politer than 당연하지 and 물론(이지). However, 당연하지 and 물론(이지) are used when you want to emphasize that sth is right. Also, 그럼, 당연하지, and 물론 are have same meaning. 당연히 is just adverb and 당연하지 is a form of 당연하다(verb). 당연히 = naturally / 당연하다 = be natural ex) 당연히 네가 그 일을 맡아야지. = Naturally, you have to take care of the job. 네가 그 일을 맡는건 당연해(당연하지) = It's natural for you to take care of the job.