Best moves and why for e4 opening?

So I was enjoy using the e4 opening quite often, as I like to push up my pawns to defend the center of the board, although I was wondering, what is the next best moves for white and why in the following situation?

As you saw, the pawn is left hanging, although often in games, players leave it undefended, is there some move to prevent it being taken, as the knight on blacks kingside can take it? I've often seen the white king castling, and defend the pawn with rook, although I don't understand why black doesn't take the pawn... Any help is much appreciated.

3) Bb5? Nd6 can give white troubles, I think there is a known trap if white blunders and if not, it is even (meaning white gave up his tempo) after NxN exN...  Black can try to keep that pawn for a little while, which can frustrate white's development and plans.

As white, defend the pawn with d6 or Nc3 is what I would do.  White can castle, black takes pawn, there is no trap here that I know of.  It can be a sort of gambit I guess, give up the pawn and black has a serious lack of development to show for the pawn. 

I have to wonder if the games where it is not taken are due to slightly different boards than this.  Nc3 for example defends the pawn quietly,  or if white had played his d pawn, the knight is pinned and white may be able to line up something tricky in other positions.

Totally unrelated, but I prefer Bc4, which gets an early line on blacks (soon to be) castled king.  Attacking the knight does not usually accomplish much, trading is unattractive and the pin has few teeth against better players.

What are you smoking?  3.Bb5 should not get a ?.  It's only White's best move here!

After 1.e4 e5 2.Nf3 Nc6 3.Bb5 Nf6, known as the Berlin Defense, White's best line is as follows:

4.O-O Nxe4 (4...Be7 is also possible, though known to be weaker at the top levels) 5.d4 Nd6 6.Bxc6 (This is clearly best, though 6.dxe5 is possible as 6...Nxb5 7.a4 traps the Knight) dxc6 7.dxe5 Nf5 8.Qxd8+ Kxd8 9.Nc3 and here Black has options.  The old old OLD line is 9...Ne7.  More common today are 9...Ke8 and 9...Bd7.  The idea is simple.  You need a favorable endgame.  These include:

1) King and Pawn (almost an auto-win)
2) Knight and Pawn (again, almost an auto-win)"
3) Knight(W) vs Dark-Squared Bishop(B) (Black's worst piece)
4) Dark-Squared Bishop Ending
5) Pure Rook Ending

And the following are most favorable for Black:

1) Dark-Squared Bishop(W) vs Knight(B)
2) Opposite Colored Bishops

And the most "neutral" ending is:

1) Knight vs Light-Squared Bishop

The middlegame is all about fighting for g4, a light square.  If White can't get in g4, it's very hard to get in f5 as the e-pawn is already on e5 (unlike in the Exchange Ruy Lopez, where it's on e4).  The worst trade you can allow as White is Black's Dark-Squared Bishop for your Knight, like if White makes the bonehead move Re1, and allows Bb4, pinning the c3-Knight to the Rook.