There's nothing at this time to say that Kirk served under him. It's possible that there is a Enterprise Captain between Pike and Kirk that is killed.
Technically not allowed by "The Menagerie", in which Kirk explicitly says he got the ship from Pike... If Kirk pried her out of the dead hands of some other predecessor, why would Pike be there to do the formal handing-over?
He means "about your age." It's just an inconsistency in the script.
No script could plausibly have such an inconsistency. It's a massive story point that Pike preceded Kirk in just about everything, that Pike used to be what Kirk is now. The writer would never have fumbled this particular ball.
It's also a bit unlikely that the actor could have fumbled, and the director and everybody reviewing the shots would have missed it before the episode went on air.
Not sure whether mentioned explicitly anywhere, but I thought that Kirk was supposed to be the youngest Starship Captain.
Nope, not mentioned in any episode or movie. And it's unclear whether Roddenberry or the writers intended such a thing, beyond a few early ideas.
As for the Fleet Captain thing, it could be that this is a rank higher than Captain, in which case Pike would once again precede Kirk. But it's also possible that Fleet Captain is the very same thing as Captain, just a fancier way of saying it. After all, "Captain" is ambiguous because it can refer both to rank and to position. Perhaps when it's important to make the distinction between those two, the rank is called "Fleet Captain"? That is, Pike would previously have been captain of the
Enterprise at Commander rank, but would at this juncture have been promoted to be captain at Captain rank, which is confusing unless specific language is used.
Timo Saloniemi