To answer your questions:
1. That's because d4 is the strongest move by far, followed by Bd4. A prophilactic move like h3 is not necessary and overly defensive. If you play 1. e4 you're expected to be agressive.
2. Since your opponent played h6, playing d4 is still possible.
3. Playing d5 was a mistake. You transitioned into a type of KID in which black gets an easy and even unstoppable f5 pawn break, while your queenside attack hasn't even begun yet. By playing d5 I would say the black now has the advantage.
Furthermore, black's little maneuver put his queen in an ackward square, blocking the DS bisshop. So you'd like to open up the game because bisshops do better in an open game. By taking on e5 you also effectively make the queen the only defender of the e5 pawn. You'd be at an advantage since Nf6 can easily be answered with Bb5 and black has issues with his development.
4. Since black has already started his kingside assault, you should've gotten the knight out of there and block the kingside as much as possible. I wouldn't have castled kingside either, way too risky.
5. I would seriously question your idea to open the e-file, because black has both a stronger central and kingside presence. At that point you should've considered closing up the kingside, maybe by giving him a passed pawn and then blocking it with a knight, and then attack on the queenside.