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I have a bit of a habit of playing 1.e4 with the follow-up 2.f4 in lightning (1 0) games.
In response to 1..e5, it becomes the King's Gambit. Apparently, against 1..c5 it's known as the McDonnell Gambit and against 1..d5 it's the Williams Gambit.
I'd like to know if there are particular responses in Black's first two moves which mean that proceeding blindly with this kind of attack is a bad idea, and to slow down and play more cautiously instead?
If you can guarantee that your opponent isn't going to play the Scandinavian (1...d5) or Alekhine's Defence (1...Nf6) then you might get away with it but either one of those replies by black is going to leave your pre-moved 2.f4 giving you a roughly one pawn disadvantage. So, not a good idea.
I don't play 1...e5 anymore but when I did I played 2...d5 (Falkbeer) following a Morphy game from back in the day.
Also, Fischer published a refutation to the entire line. You might want to look into that.
Playing e4 followed by f4 can be dangerous, as black can easily and most likely will play f5 or e5 and capture your pawn, and never premove e4 f4, always proceed captiously in openings. Make sure your opponent does not play Nf6 and pick up your free pawn though!
I start by answering your question.
[FEN ""]
1.e4 d5 2.f4 dxe4
Looks terrible for White.
[FEN ""]
1.e4 Nf6 2.f4 Nxe4
Might cause Black to use more time to play, e.g. after 3.Nc3.
Now, if you want to play quickly in the opening phase, there are better ways. For example 1.Nf3, 2.g3, 3.Bg2, 4.O-O.
Your King is safe and you can start building up an attack.
I think in 1+0 bullet, you want to reach a middlegame quickly and then hope your opponent blunders there. A safe king is worth a lot, because the player with an unsafe king will spend more time double checking against threats.
Black has many good responses as given by others. Against c5, I'd say f4 is a too early commitment by White. For example 2...d5 by Black and if e and d pawns get exchanged one way or another then that f pawn looks stupid, having weakened both diagonals leading to f2 (allowing tactical motifs even after a short castle), and having blocked the natural development squares for the dark bishop.
If instead White pushes e5, then Nc6 and g6 both look good. Notice f5 is also available for Black's Bisop or Knight. White's e5 wedge is often used as a tool to launch kingside attacks but it is not easy to transfer in the queenside pieces to speed up such an offensive, which should allow Black enough time to push their own agenda.
Not a lot of people play the Alekhine defense, especially in bullet. so not a lot of dangerous moves for Black, except 1...d5 but that is not really dangerous provided you do not pre-move 2.f4?! but even then, there's a chance that your opponent also pre-moves 2...Nf6?! to which you can respond with 3.e5!.
Nothing comes without a price. if you want those extra milliseconds then you have to accept the fact that once upon a time you will get in trouble if someone plays the dreaded 1...Nf6 or 1...d5.
Best of luck!
