Quick question:
The global Inbox isn’t syncing via DTGO?
Why? All other databases are working fine. I’ve already uninstalled DTGO => installed it on my iPhone and iPad, but it didn’t help. I had the database repaired (no idea if that makes sense).
But it’s only the global inbox that isn’t working?
Do you mean “Inbox”? Have you actually turned on (by the blue switch) sync for “Inbox” in the sync settings?
What version of DEVONthink ToGo and iOS are you running? Up to date? What third party sync service or method are you using?
FYI, reinstalling software like you did rarely fixes issues …
Yes, Inbox 
Translation fail
Everything ist uptodate an with newest Version. Sync with Bonjour. Yesterday everything was fine
Is anything logged to the Window > Log panel?
No.
And a verification of the global inbox (see File > Verify & Repair Database…) is successful? Does rebooting both the Mac and the iPhone/iPad make a difference?