Q and K vs Q, K and anything except a R or a Q extra or 2 pawns extra is a draw.
(i might be wrong i'm basing that on experience thru my games)
KP endgame
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What if on the first move if instead of capturing the pawn white plays Kf3 instead?
Kf3 hxg4 hxg4 fxg4 Kxg4 Kf6
Each would have a passed pawn. Could either side advance?
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1...h5 is a blunder. White wins easily after that because he will exchange all the K-side pawns and win with the remaining Q-side pawns. Here is a short line 2. gxh5 Kf6 3. Kf3 and Black can only choose which way he wants to go down. If Kg7 White plays e4 and if Ke5, then Kg3 and h4.
Good drawing chances are in the line 1...Ke5 2. Kd3 Kf6 3. Kd4 Kg6 5. Kd5 h5 6. gxh5+ Kxh5 7. e4 f4 8. e5 f3 9. e6 f2 10. e7 f1Q 11. e8Q+ Kh4 12. Qh8+ Kg3 13. Qh6 Qf5+ with a likely draw despite the pawn deficit.
Your line with 2...f4 is much weaker and only spoils Black's chances to hold the game.
Nytaclop is right about white's idea line of play following h5. However, black can draw pretty easily with Kf6 or Ke5. The general idea being that black has two threats. The first one is that he can win the e-pawn by taking it. The second is that if the white king gets too far away from the kingside, black can easily force a passer... So long as black keeps both these threats active, the white king is tied up defending against both of them and the game's drawn.
1...h5 is crazy :) but some others pointed that out already.
In the 1...Ke5 line, black shouldn't break the tension for anything (except a passed pawn if white's king leaves the area), so 2...Kf6 or even 2...Kd5 look much better.
I saw a King and Pawn endgame played by one of my friends (playing White) at an OTB tournament this weekend. As he played it, I went to skittles and analyzed it with a couple of others. We came to the conclusion that Black has a draw. Were we right? In the actual game, Black lost.