1. Kingside castling is more common because for one thing it's a lot quicker. You only have to move 2 pieces whereas on the queenside you need to move 3 and often want to move your king to b1 afterwards.
2. Generally you don't want to have moved any of the pawns. It creates a target for attack. e.g. if you have a pawn on h3 black can sometimes sacrifice a bishop there to open up your king's cover, or push a pawn down the g file to open up lines against your king. This 'lever' isn't available if your pawns haven't moved.
That said, there are benefits to h3. It protects against back rank mates and can stop black's pieces using g4 (often a bishop pinning your knight). So for example in the Ruy Lopez white may play h3 to stop a bishop pinning the knight so the knight can support a d4 push.
kingside or queenside castling with moved side pawns?