Do ENTJs feel the need to prove they are better than the others?
Do they think they are the best anyway?
Do they see the others trying to be the best as annoying?
Do they sometimes feel inferior to other types, and if yes, why?
I'm not even sure what better and best mean in this context.

Questions 1) and 3) go together. If we're dealing with something like a music audition, then of course, competition is the nature of the activity. Outperforming others here would be the criterion of success. However, if we're dealing with an activity where teamwork is required, one can demoralize an entire group quickly if people don't feel valued. There is a time and a place for competition. My answer would be: the ends in question determine the appropriate means.

Question 2) is empirical. If someone is better than me, that's an opportunity to assimilate what they know or what they can do. For example, I can quickly find out if I know more about Kant than another person by talking with them and asking pointed questions. The same holds in any scenario. You'll excel with this attitude whether you like it or not.

Question 4) is downright retarded. I don't feel inferior to other types, and I don't feel a sense of allegiance with the ENTJ type either.