1- definitely Luke's kid

2- Yes it does. If it didn't wipe out the planets and just left them to freeze, that would give the enemy time to escape.

3- I didn't get the impression Kylo killed the Jedi, just that Luke stopped training.

4- I think it's just another name for those using the dark side of the force. We may learn more about them in Ep8 but I don't see a significant difference between them and the Sith.