What happens if I want to play a New Rule but there’s already a New Rule in the center on the table? Does it replace the old New Rule, or do they peacefully co-exist?

If they contradict
It all depends on contradictions. There’s no limit on the number of New Rules that can be in play at once, as long as they don’t interfere with each other. If the New Rule you are adding has nothing to do with the existing New Rule, both rules remain. However, if both rules are about the same thing, for example how many cards to Draw, then the new card becomes the rule and the old rule is discarded.