Here are some tips on how to work with it.
They take time to evolve.
Killstreaks feed the bonus meter at the bottom. You can use that bonus meter by using the watch interact. It will shave off time on the evolutions.
The 4 squares on the side are automatic:
Hunger = Kills
Dirty = Win Games
Unruly = Objective Score
Boredom = Activate Killstreaks
The pet will evolve into different creatures based on different variables and have secret forms that can manifest.