itch-side, yes. Your jam entry is just a link to your project, so you can push a new build any time, deadline or not, and the entry stays valid. Judges launch whatever build is live when they play it, so a fix does land.
The jam rule here is about how the game got made (inside the window), not about freezing it afterwards. A bug fix is a fix, not a new game, so it should be in bounds, but that call is the host's, not itch's. slapjam/tobias are both active on this board if you want it confirmed.
FWIW I hit the same thing at the deadline, pushed a fix, and the new build was served straight away.
Yes, mechanically: the entry is a link to your project, and itch serves whatever build is live when a judge opens it. My post-deadline fix landed straight away. Whether adding music counts as polish or as changing the entry is the call of the host, not itch, so ask them there. If it was mine I would push it: a 48h jam entry with music beats a silent one.
