Is there a rejection cooldown for moderation? I was rejecting runs that had unavailable videos and suddenly it told me that I "don't have permission to reject this speedrun." or something along those lines.

This isn't something I've personally heard of or encountered, but I wouldn't be surprised if most actions had some kind of soft rate limit to prevent people from overwhelming the servers and/or to mitigate malicious behaviour. Or maybe there was just some unrelated server hiccup while you were doing that. Who knows. Either way, I wouldn't worry about it too much, just try again later. If it's still doesn't let you later then it's probably a bug.