Photos recognising faces and pets

I need to have photos open but minimise it, or fully quit the app (?)

Either one should suffice. Just do not use Photos. Keep your Mac turned on and photos running over night, when you do not need to use your Mac. If your photos are syncing with iCloud Photos, ensure that your Mac is connected to the internet, so Photos can download the originals.

If your Photos Library is large, it may take several days before you will be seeing the first detected faces.

With that library size the scanning for faces will take a very long time, even on a Mac with a very fast processor and lots of memory. o not interrupt the scanning by attempt to repair the library or changing the iCloud settings. It will just take longer, because Photos will have to start over from scratch.

1. The grey squares may be synced faces from your iPhone and should resolve itself. I am not sure what you mean by 50/50 split?
2. Each device is scanning for faces on its own, because the face detection algorithm is specific for each system version. But the faces names synced from your other devices will be used to name the detected faces, if they have been detected in the same position. You may want to use the command "Confirm additional faces" occasionally for each People album, to confirm the names for the faces that have been detected in a slightly shifted position on both devices.