First, let's get the nomenclature right. This is a "free kick after a fair catch". All kickoffs (to start a half, after a TD, FG or safety) are also "free kicks". I assume we are not discussing those.
1. A free kick may be used after a fair catch or awarded fair catch
2. A kicking tee is allowed for a free kick. A block is also allowed for a free kick.
3. The ball does not need to be snapped....basically a team could put their kickoff team out on the field with a kicking tee and if the ball goes through the upright then we have 3 pts.
4. A free kick is just like a punt. Should the ball not go into the endzone then the ball can be returned
5. The defense must be back 10 yards and cannot cross that 10 yard area until the ball has been kicked
All true
6. The kicking team cannot do anything else besides a kick (example - no passes or runs...etc.)
I'm going to make an assumption here and say that you mean once they declare they are taking a free kick after their fair catch. If that's true, then yes, I agree. Obviously, they could always run a scrimmage play, but not after declaring the free kick attempt.
7. if the fair catch takes place at the 30 yard line.....the free kick can be kicked from the 30 and the remaining offensive players are allowed behind the kicker just like a kickoff. Example - remaining players are at 34
The remaining players MUST be behind the 30, and no further back than the 35.
8. If a team fair catches with no time left on the clock then game would be over and no free kick allowed. There are no penalties on this play.
Two sentences here, one is correct, the other is not. You are right about the clock. But there can be penalties on the play. Either team could encroach, there could be illegal blocks, etc after the kick.
One thing not in your comprehensive list: the free kick can come from any point between the hashes on the line where the fair catch was made or awarded.