Why not 4... Kf8? I'm sure there's a reason, I'm just not seeing it.
A endgame pattern to know.

Why not 4... Kf8? I'm sure there's a reason, I'm just not seeing it.
If 4... Kf8 then 5. Qd8#!

I think there's multiple ways to do this after the 5th move. For instance, I tried 6. Qf4+ to get the pawn with check. Is this incorrect (as in, a slower mate)?

I think there's multiple ways to do this after the 5th move. For instance, I tried 6. Qf4+ to get the pawn with check. Is this incorrect (as in, a slower mate)?
No that is fine too. The mate then most likely ends at h8 as a support mate.
Today GM Polgar posted a pattern. I wanted to share it with my friends here. There comes a point that you can make many different choices I offer one variation. The options come after move 5.