Do your companions care how you act?

I know that your companions will gain favor with you if you accept or complete a quest that matters to them.

But if I go around and steal some caps and they are with me, will they care?

What other things should I do or NOT do depending on what companion I have following me?

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.

Very much, yes. A full chart of what each companion loves/likes/dislikes/hates can be found on the Fallout 4 Companion wikia page.

It's worth noting that a companion does not even need to be your active companion to like/dislike what you are doing if you are around them. It is possible (I do this to efficiently get the perks very early on) to increase your relationship with multiple companions by sending them to the same settlement and carrying out actions that they all approve of. (Danse, Garvey, and Codsworth all love when you modify/build armor and weaponry)