First, don’t put any of the database on a sync folder. If you have that, it’s not gonna work and probably will break a lot of stuff. If that is what you have done, don’t. Move to the local drive.
Oh, I now understand that the new database is showing up on the Local list but is not synching. I previously erroneously understood it to be missing from the Locals list. On the Mac, for the Apple CloudKit sync location, check for that database the “Last Synchoronisation” date/time and compare. I’m guessing it was copied up to the sync location (unless Apple’s CloudKit is mis-behaving).
On the other devices, for the Apple CloudKit sync location check that sync is turned on.
Get Bonjour working. Details in the DEVONthink Manual with supplementary info at Bonjour Simplified
Yes, you can have multiple sync locations. If this were not possible, then DEVONthink would have disabled additional locations. 
But, for a database, only use one sync location at a time.