Do factions and companions care what other factions you associate with?

As far as joining goes, you can join all of them without any problem, at least at first. I'm a member of all factions and no companion complained so far.

I even took Nick Valentine with me when visiting the Brotherhood, which results in a few comments but nothing more.

At some point there are consequences for doing certain missions with a faction, but I stopped before the point where I have to decide and avoided it so far. So I can't tell anything more about the late game. But you can certainly join all the factions at first without worrying about your companions.

Yes, you can join any faction while belonging to any other factions.

There is however a point where you have to select loyalties and then your companions will care.

For example, I went with the Institute, which resulted in destroying the Railroad and the BoS. After than I returned to Red Rocket Truck Stop, where all my companions where and I was attacked by Danse (BoS) and Deacon (Railroad).

The others did not care or react.