I've driven the Khans away, solved the problems at Omerta and White Glove and now I have BoS left.

I did two quests for the Elder, and during the last one with the air filtration thing I decided to help the Head Paladin take over.
Now, what do I do from here? I would like to get on good terms with BoS if possible but I'm unsure of what I have to do,

Also the door to the bunker is locked by key, so I did the second quest by using the Movetoqt command and now it's still locked.
Will it open after a few days when the Head Paladin has taken over or will they come out?

I'm guessing I have to wait a few days for the Head Paladin to take over, as he said, and then I'll be able to talk to him further, us this correct?
I'd prefer not to ignore them during the Wild Cards quest.

You can't be on good terms with the BOS, sadly. Not in a mutually beneficial way, at least.

For side-bets all you can do is ignore them or wipe them out.