I'd like to join the Kings, but if I do, he won't stop hostilities with the NCR. I don't really mind that
but will I still somehow be able to activate the NCR quests and do some of them before I go
independent? And another question, I already went indepenent once, but I think I made the NCR
leave in any case so hostilities don't really matter anyway, right? Thanks!

The King will only give you one favor but you get 3 or 4 choices. You can still end hostilities between Freeside and the NCR, but with a less diplomatic approach.

As for joining the Kings, i was able to join the BoS and wear Power Armor in front of the NCR, and they still accepted my help. I'm sure joining the Kings won't hurt your rep with the NCR.

But later in the game if you continue with mr.house quest, you will be asked to stop hostilities between the kings and NCR , so if you already took your favor with the king you won't be able to finilaze peacefully the mission you will have to kill someone else... so wait until you get to that point and use your favor wisely! cause being part of the kings doesn't seem too get you that much in return. Iknow I did it for kicks and beside the prestige of being one of them it's useless unless you want one of their costumes and hair cut.

But later in the game if you continue with mr.house quest, you will be asked to stop hostilities between the kings and NCR , so if you already took your favor with the king you won't be able to finilaze peacefully the mission you will have to kill someone else... so wait until you get to that point and use your favor wisely! cause being part of the kings doesn't seem too get you that much in return. Iknow I did it for kicks and beside the prestige of being one of them it's useless unless you want one of their costumes and hair cut.