Not that it really matters, but...
Ultimate is the name of the game - one word.
Frisbee is a brand, and not a particularly popular one among ultimate players.
FWIW - most serious players I know actually prefer Discraft as their disk of choice.
Yes, but if you want to discuss it with someone who is unfamiliar with the term, I think that is the quickest way to get the idea across. Even then, they get it confused with disc golf.