Kind of a dumb question, but I'd like to release a patch for my game even though it's past the deadline. Is it legal for me to do that or should I just wait? Or could I release it as a separate download from the original version?

I've personally released patches for bugs since I don't want people to come away with a bad impression of my work.

I had to fix a game breaking bug, the game wouldn't even load after some problem happened with our server. I guess that as long as you don't change too much stuff it isn't that bad.

Since the rule of the jam didn't mention about not allowing bug fixes, I would say it is okay to do bug fix.

Even Ludum Dare permits that.

As long you don't add something new things/feature to the game, then it is okay.

This jam doesn't lock your uploads so you're free to do what you want. If you do create a new build outside the scope of the jam I recommend you label the files and leave the old builds around during the voting period so people can still access them.