I recently made a run and the only proof of it is in my Hypixel replay while I was nicked. Am I allowed to still submit? Is there a way to take the nick off the replay? I tried turning nick hider on and unnicking in Hypixel, but it won't change.

There is no way to disable a nick in a replay. It is fine if you are nicked, and so yes you are still allowed to submit. Just make sure to include you going into the replay from your '/games' menu in the lobby, to prove it is your perspective