As part of a longer sentence, then we usually use the first one (according to the rule cited by Amirreza), for example, "I will definitely do it". As a stand-alone short-answer, we usually use the second one, although in American English, we mostly just say "I definitely will", with "do" understood. There is also a frequently-used short phrase (not a full clause, not a sentence), "will do", which is often used to indicate the speaker will carry out the task that was just assigned.

The first sentence is not complete. You can say that “I will definitely/certainly do something.” And “I definitely will./Of course I will.” are more common expressions.

The way I see it, 1 is possible but not very common 2 is ok if you drop the "do" => "We are going to the cinema tomorrow night, will you join us? - Yes, I definitely will."

The first one should be I will definitely do something. The second one is fine and quite common.

Only the first one. Adverbs should go after modals.