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.