Can a player who is not in my community join my community's game or pay for a friend?

Yes. Community membership is never checked when someone joins a game, including games linked to your community. Anyone who has the game link can join, and a player can bring a friend who is not a member.

What is checked for every player, friends included: your bans (a game is covered by the bans of the person who created it) and the game’s participation requirements (approved photo, age, rating, contacts). If one friend in a group join is banned or fails a requirement, the whole group join fails.

To keep the game to your own people, make it “Private” and share the link only in your community; see “How do I make only my own players sign up for my game?”.