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 answer only on behalf of myself.
1)
I'm interesting of improving myself, and I usually try to establish the basic stuff first, and expand on that. I discuss a lot about what's the best methods, best points of view and best interpretations on anything.
In the beginning of a learning process, everything is open. Soon after that - and usually - only the more advanced part of the theory are open for discussion; I'm usually ready to point out that I've already established the more simple parts of the theory, so I don't want to talk about them unless I see strong evidence to change it.
So, I like to communicate where I'm at. Sometimes, when I want to direct the course of discussion to more fruitful areas for me, I just say bluntly that these ideas X are unsophisticated and I've already resolved that, let's look at Y. The point is not on my superiority, though. I might know that other persons might reach the same goals with different starting points, but as I have invested time and effort to get where I'm at, I'm not that interested of different starting points anymore.
Hence I say: I've got that all covered, why don't we talk about the more advanced points now.
This is not a point that the other person is inferior, or their opinions might never lead to anything. It's just that I personally feel being closer to the solution in my situation with what I have.
2) No.
On the other hand, some things I do require me to be the best of 10 people in thing X, other things require me to win 99 out of 100 people, others even more. We all specialize. A painter who's only better than 90% of the population in painting isn't going to make a living of it. Many averagely paid job positions are filled with people who are better than 99,9% of people in their selected thing. This is specialization. So, I understand that I'm the best in an average, randomly selected sample of size N in skill Y, but that doesn't make me the best alltogether. Other people have their areas of expertise, and they likewise win me in their selected skills and abilities. Being best in something - in a limited fashion - is quite a usual thing.
3) Yes.
I see it annoying for someone to claim superiority in something which I see as situational, relative, not useful for my situation, not required for them, or if I see it as an ego trip.
4) Yeah, constantly, in some aspects. It comes with the understanding of being good in something. For example, I lack some kind of "kinship" or "group spirit" of a football team or something like that; I find no joy at all from the kind of bonding in such environments. Consequently, I totally suck at the kind of team work like that - while I excel at other kinds of team work. I see that in order to get a job where my colleagues are team players, I must be technically very good, in the top 5%, to be able to deliver. I understand I'm filled with deficiencies like this, and because I'm able to get in the top of my selected competencies, I go there, and then I'm about on equal par with everyone else to whom I'm comparing myself against.
My tendency to compare is primarily a result from trying to earn my living. It fills my life and spreads to other areas of life, as well.