Nope! We don't have such a requirement. Besides, in most countries copyright is automatic. In plain English, you already own any game you make. Big companies only register copyright so they can sue others for infringement more easily.
That said, you should make sure you have the right to publish any third-party material you use. E.g. art made by someone else. That means respecting the license if it's open source, or getting permission / buying a license otherwise. If you don't do that, people might complain, and our admins would have to take action.
Trademarks are more complicated (and expensive). My personal, amateur advice is to do a search before picking a name for your game, to make sure it's not too similar to others. That can be tricky, and it's not much, but it's something.