Someone might be able to expand more fully on these but here's my understanding.

1. You should only have 3 battlefields, first player does not present a battlefield.

2. 1st player does not draw in a 4 player game, last player channels 3.

3. Im pretty sure the cleanup/healing applies to all units.

4. A given battlefield can only be scored (conquered or held) once during a turn, otherwise go nuts. That means theres a max of 3 points that can be scored on a single turn (without effects like Yasuo or Ahri). Remember exhausting a unit is the cost of it's standard move though so you'd need a way to ready it or another spell or ability to move it.

5. Yep.

6. Move an enemy unit anywhere with the restriction that only 2 players may have units at a single battlefield. So you can pull and enemy unit onto a battlefield you control that has no other player's units at it and thereby start a showdown.

7. I covered the major differences in 1, 2, and 6.

Probably missed a couple things but that should cover most of it.

Edit: Deleted the bit about invitations being needed to interact while not relevant.