When I Clip items in DTTG sometimes they do not show up the same day in DT3.
When you clip items how? If you’re referring to using the Sharing extension to clip from mobile Safari to DEVONthink To Go, that would be highly unusual
Do I need to open DTTG and wait for the entire sync to complete after I clip something?
Wait for what entire sync" ? Are you trying to clip into a database that hasn’t yet been imported into DEVONthink To Go?
Can I open DTTG and then switch to some other app or turn off the phone display and have it sync in the background?
DTTG should be in the foreground and the mobile device awake for the initial sync. The Background App Refresh option is controlled entirely by iOS and only allows approximately a 30 second window, when it allows it to happen. This means we can’t control if, when, or how long it happens. Subsequent syncs are faster since there’s less data being transferred.