Are there any non-coin tournaments I can join? I see I can create them, but I don't see any without coins. Or can someone invite me to a couple?

Coin games are public (= Everyone can see them) all other tournies are private and you need either the link to it or an invitation.

How do you get an invitation? Join a clan or become famous and you will be spammed with invites.

lol, "just become famous" - easier said than done.

How do I join a clan? Is that unlocked at a certain level?

^ No you could ask to join a clan at any time you want