So far, the only attention I've paid to my companions in adventure mode is to make sure they aren't killed. Is there anything else about them that I should monitor and care about, e.g. hunger, thirst, sleepiness, equipment?
Not that I am aware of. I don't think anyone really cares if they get killed or live either. I just use them as moving meat shields when I play with them.
Also you can take and use their equipment after they die, so letting them die is actually beneficial in the early stages.
I'm pretty sure they don't eat or sleep for some reason. I once was giving them my food in case they got hungry, until I ran out and started starving. I killed them for the 6 prepared fish they'd accumulated in their inventory.