I'm standing on a staircase, surrounded by hostiles, with my pets just outside the hostiles. They are helping, but I'm getting low on HP and need to go up to restore. It's going to take a while, and I can't bring my pets with me. Will they continue the fight (possibly, winning or dying)? Also, might a pets hunger influence its will to fight on?
For the most part, Time is paused when you are not on a level. (Source 1)
Your pets should only follow you upstairs if they are adjacent to your position when you use the stairs. They will slowly grow feral if you leave them behind on a different level. Your pets only need food to survive, but they should feast on the corpses of enemies that are made of meat. (Source 2) Corpses will still rot when you are not on a level, however, and since creatures are frozen in place, your pet probably won't be eating anything while you're gone. (Source 3)
The tameness of a pet depends on how well your pet is treated and fed, and will be affected by how long you spend on another level. Detailed information may be found here: (Source 4)