The 10814 error code you're getting for dbapi-3 is expected when no application is installed to handle that URL scheme. That would be expected, for example, if the Dropbox app isn't installed (e.g., on the simulator). Is that the case in your scenario? (The Dropbox Chooser is actually built in to the official Dropbox app, so the official app needs to be installed for the Chooser to work.)
The "This app is not allowed to query for scheme" error you're getting for dbapi-1 would be expected if you're not registered for dbapi-1 under LSApplicationQueriesSchemes in your app's plist. You mentioned you did that, but can you double check that was entered correctly? If it's not working as expected, please share your plist so we can try to reproduce the issue.