Do your companions care how you act?

Yes

For a simple example, go to Sanctuary and ask Codsworth to be your companion. Use the flare gun to attract a couple killable NPCs and throw a grenade at them. The message "Codsworth hated that" should appear on the screen.

Moreover, you're able to ask the companion directly about their feelings on your relationship with them. After committing enough suitably negative acts, Codsworth in particular will comment that he "barely recognizes you as the man you once were," or something to that effect.

The short answer, at least until someone with better sources than I comes along is: yes, your companions do care.

I know at least Piper dislikes it if you try to weasel money/rewards out of people through the yellow/orange/red persuasion options. She prefers it if you just offer to help people without badgering them for more stuff.

But, the only companions I know are the dog and Piper. You should be able to get an intuitive sense about what your current companion is after. And, of course, the game will tell you when they like/dislike something - you should be able to figure out what it is they don't like about what you did.