I know it says in the rules that you can't do covers, but it would it be O.K. to post covers if you made sure you didn't put your name on it or enter it in the battle?

what has not putting your name on it got to do with no covers being allowed?

not allowed, although personally I don't mind them

Be careful.. respect the copyrights of the owners of these rights. If someone gives the evidence of authorisation to publish his cover, so why not? Its a basic question and I personally would not allow to publish any cover of my songs without my authorisation. You may play it at home for your own but if you make a cover to earn money for that I definately want to be asked before publishing.

But if we simply posted it as a remix for review without entering the battle, we would not be being paid for it at all.