The hats are for just for looks; they don't do anything special. If you're hardcore, you'll find all the hidden hats, and I mean ALL of them. Most hidden hats are obtained by achieving a statistical goal (e.g. destroying a LOT of trees).
The game controls are best explained visually, in the game. To see the keys, you can pause the game with the Return key (aka Enter).
Go Left / Right = H Key or K Key
Reverse = J Key
Jump = U Key
Cannon Up / Down = A Key or D Key
Fire Cannon = Use Space Bar
Missile = S Key