Can iPhone back up to iTunes over Wi-Fi automatically?
After consulting with Apple it turns out that 


Automatic sync should happen every time the phone plugs into the computer while iTunes is running
None of my the iTunes installations did this on macOS or Windows, which is an error
To work around this required manually syncing a folder with a photo in it to the phone, on both windows and macOS separately. After that automatic sync starts working. 


From inspection I can conclude that:


Yes
Yes, when plugged into charger
Yes
Haven’t tried yet. Probably, since backup seems to be included as a step in the syncing process. 
Backup should happen within a few seconds of plugging in the charger, five hours from the last backup the same day. Should probably happen every time, regardless of interval.