Sorry, but I have never sold a database or applied an external database on any of my forums.
When you sell databases, I'm pretty sure your selling MySQL Databases, correct?
If so, can a database from a free forum host be applied to a different forum? (If the free forum lets you view your database.)
You sell the backup of a database and all information related to that database. And yes a free forum host database can be applied but it is tricky as most of them have all their own custom edits and all that crap so whilst it can be done it's not worth it, and a free forum host will almost never allow you to get your own database.
Why would you sell it? People don't really offer a lot for DBs unless they have 100,000+ posts and even then it's not easy to sell. And it depends on what cPanel and stuff you have.