I don't see much wrong with 6. e4 instead of 6. e3.

c6 can be useful in many ways 1.It opens the a5-d8 diagonal for the queen which could move to b6, a5(with eventual switch to h5), or even c7 under circumstances from where it can support e5. 2.As you mentioned it keeps the knight away, which is not an option yet, but could be later 3.After white plays h3, black is almost forced to exchange the bishop for the knight. After white has recaptured with his bishop, the h1-a8 diagonal looks less vunerable, after c6 is played. 4. It gives black the option to play even d5 later on, and also to gain some queenside space after a6, followed by b5 Of course other moves are possible. I would never play my bishop to g4 only to exchange it for a knight, and leave it at c8, for in many kings indian defense variations, the bishop is already developed.

try to think in terms - prepares to push e5

whats wrong with Nxe5?

11. Nxe5 Nxe5 12. Qxd8 R(either)xd8 13. Bxe5 Bxe2

the idea is to play Nc5, and not to allow white to chase it away by means of b4. if white tries to prepare b4 with a3, black then simply responds with a4. e4 followed by Ne5 looks interesting here