I wrote a long series about sqlite3 and Keyboard Maestro a while back:
The beauty of using sqlite3 is that it's included in macOS, and Apple uses it extensively in their apps, so it's unlikely to vanish any time soon.
Yes, you can set their storage location (depending on how the macro is written; check out my Quick Web Search macro as an example of one that lets you specify a storage location.
But no, not on dropbox. sqlite3 is not written with multiple simultaneous access from networked machines in mind. It uses a file locking mechanism that doesn't work across a network, amongst other issues.
I don't think I want to try to address synching, given the complexities involved. But it wouldn't be hard to write a macro that made sure the database was quit on other Macs, then copied the current file from one Mac to others. But syncing changes and keeping them current both directions? Not something I'm even going to attempt :).
-rob.