Are there good variations for White against the Sicilian that do not involve an early d4? If White opens e4 and Black replies c5, that's the Sicilian defense. Typical next moves are 2. Nf3 d6 3.d4 cxd4 4. Nxd4, etc.

Are there good variations for White where he DOESN'T play d4?

I've heard of 3. b4 (Wing Gambit), but I don't like gambits.

Does it make sense for White to play d3 instead, either before or after Bc4?
It's quite common the Closed Sicilian, with, for example:

[FEN ""]
1.e4 c5 2.Nc3 Nc6 3.g3 g6 4.Bg2 Bg7 5.d3 d6


The Grand Prix Attack is of course another possible way to face any Black player after 1.e4 c5. Nowadays the main line is:

[FEN ""]
1.e4 c5 2.Nc3 Nc6 3.f4 g6 4.Nf3 Bg7


After 1.e4 c5 2.Nf3 (any) 3.g3 White can build a King's Indian Attack, which is less common and less sharp, but perfectly playable.

These variations are not as popular as the "Open" ones, but you may encounter them from time to time even during high-rated torunaments.