I've got quite a collection of Unity assets, some of which are not free. Is it allowed to use them? According to the rules, I'm allowed to use the assets that I have a legal right to use. As I have it, it seems to be alright. But I wanted to clarify it to be sure.

Yep! The rule is targeted at people taking assets from commercial games without permission, and things like that. If you have purchased assets and are licensed to use it in a game, you can use it in this game jam.