1. e4 e6 2. Bc4

I've seen the following many times.

In the past I've always just played the simple c6 to allow Bd6.  Now I'm wondering about d4 as an alternative.  I once read that 1. d4 Nf6 2. d5 is generally considered inferior for white, so this is something I worry about.  The two positions are very different, so I figure it may not be a bad option here.

Here is another one I see on occasion, so I'm wondering about d4 here.  With Qf3, I typically expect someone who rushes to mate on f7, so this looks like a reasonable continuation.

There seems to be some complications along the a8-h1 diagonals.

With your pawn so solidly on d5, the bishop looks silly. What's it doing there?

But then you voluntarily move your pawn and suddenly it's the best bishop ever with a clear line to f7, no black pawns that can go in between and no way it can be attacked. And it's white's move.

I dislike the idea of 4...d4.

That's a good point.

well, this is the same idea as in the the sicilian: why does white simply waste time so early playing a move like Bc4, there are certain exceptions of course, but here, 2.Bc4?! gives black full equality, and i like white's stinky bishop.

1 e4 e6 2 Bc4 d5 3 exd5 exd5 4 Bb3 What is to be black's best move here 

I'd say 4 ... Nf6 5 d4, Bd6. You've got both of Black's Bishops aiming at the White Kingside, and you could throw in a5 at some point gaining space on the Queenside and threatening to trap White's Bishop.

Stockfish already gives Black a quater to a third of a pawn advantage.