u can find this on books...
The idea behind the opening
Uh. See, that's a problem with teaching openings, especially complex openings like the King's Indian, to beginners. Actually the King's Indian does not go like this. If white combats the King's Indian by choosing the Classical, and black in return chooses the mainline there, you are likely to arrive at move 9.
9.b4 is indeed the most popular move at the moment. In the late eighties / early nineties it was 9.Nd2. Before that it was 9.Ne1. Of the alternatives that you suggest, only 9.Bd2 isn't outright bad, 9.Re1, 9.Bd3 and 9.a3 really have nothing to do with the position and might as well have been produced by a random move generator.
As Poucin pointed out, there is an endless amount of books on this subject. I recently bought one from Kotronias' series. But on your level, it's a much better idea to study tactics. And if you want to study strategy, then it's a much better idea to learn about pawn structures in general.
There's this guy in a chess club where I give monthly lessons, he isn't even a terribly bad player (FIDE maybe 1750, when he was younger 100-200 points more), he's been playing the King's Indian and the KIA all his life, and he firmly believes that "the idea of this opening is to play f5-f4, attack the kingside and sacrifice the bishop on h3". That is so sad to see and hear, because it shows how his whole approach to chess (which he contributed a huge amount of lifetime to) was completely wrong, and he never got a chance to understand what he's doing at the chess board.

Exception for bullet. There's no a specific idea on bullet's opening. Checkout my examples here:
https://www.chess.com/forum/view/game-showcase/special-dedicated-to-my-angsa-cantik?page=6

Rumo, what you mentioned about the player who plays the KIA all of his life and still does not have some basic knowledge...
This happens with many players who are stuck at the same rating for a long period of time.
I see the same thing in duplicate bridge. Players who make the same mistakes over and over again and year after year after year. If they were open to learning something different they could greatly improve their game...
I do very little study of openings. I really don't see the point in it. Sure I know the moves of a lot of different openings and defenses but that stuff is only moderately useful if my opponents play along. Most of my opening "study" involves traps and funky openings like the albin counter gambit. I study stuff like that so I don't fall prey to cheap tricks in blitz. Y'all ain't getting me with any lasker trap nonsense!
Most chess players know the three most important things to do when a game begins; develope pieces, fight for the center and castle. But by move, say, eight I find myself thinking: "Hmm, where do I put my last two undeveloped pieces? What is my plan from here? What are the idea behind this opening?"
I recently attended an opening lecture given by an IM. He told us about the King's Indian Defence (KID) and the idea behind the opening. After learning this it's a lot easlier to play these games.
The KID goes like this:
White's idea is to play on the queen side and at some point play the pawn break c5, exchanging the c-pawns, opening the c-file and giving Black a weak d6 pawn. Black's idea is to attack White's king by advancing his king side pawns. The game could continue:
Do you know anywhere I could find lectures like this one? Where the idea behind the opening is expained at beginner plus level? I'd like to know about the plan for White and the plan for Black and the key issues/squares/pieces in some of the more common openings.