1.e4 e5 2.Nf3 Bc5 3.Nxe5 Bxf2+ 4. kxf2 attack


Well, for one thing, you can play 3. Nc3 if you don't want to take the pawn. Honestly, you are fine in that position - you can simply play g3. But if you don't like the sac, then play 3. Nc3 to protect e4 so that when he does take and plays Qh4+, e4 is protected.

I guess I don't understand why you want to stop the attack - you seemed to get a good handle on the position.

Google search "scholar's mate" and you'll see a lot of material about this type/style of attack. It consists of playing out the bishop and queen early to attack the weak f7/f2 squares (defended initially by only the king). Sometimes you might see the knight in the attack too.
Ways to prevent this are to keep your knight on f3 to defend the h4 square against the queen, or advancing the g-pawn to g3 (with the intention of a fianchetto).
With important squares, it's helpful to count the number of attackers vs the number of defenders. In this case here, you have two attackers (queen and bishop) against one defender (king) so the square is lost if you don't do something about it.

It is actually 2 pawns for the bishop, but yes, I agree totally with this line you posted.
It's one pawn for the bishop since you have already captured on e5.

Some people are simply afraid of their king being "exposed", but in all reality, Black must prove himself in these lines. Sorry, you are right - White wins the e-pawn. 1 pawn for a bishop, in exchange for a king that has plenty of room to get away, plus tempos he will gain on the queen.
I have a friend who plays this line. White must simply be careful for a handful of moves - after that, he is just fine!
This is how I would play it. Though I didn't think anyone bothered with this line, just like 1. e4 e5 2. Nf3 f5.