1) How many air conditioners have you set up with IHC? I ask because you mentioned in 3) rooms, so I’m assuming there’s multiple air conditioners, which leads me to think your Google Home may be conflicted. One suggestion- change the name of your air conditioner in the masterbate room to Pikachu or Snorlax, some unique name, and test. It should work.
2) No you don’t, Google Assistant app on your mobile devices would suffice.
3) Assuming your air conditioner remotes are IR, refer to 1)
4) It’s normal, the local devices here refers to Google Home Mini, basically Google Home devices. You don’t have any, so it shows as zero perfectly fine.