You might want to look at alternative solutions.
First, dropbox is a decent way to share files, but at best would be for one person sharing across multiple locations. I don't even know where to begin with the conflicts that you're going to run into trying to use dropbox as your source management system. Look into git or svn. svn is pretty user friendly with tortoise on windows, and there are places that offer free repos.
As for mysql, trust people here when they say you don't want to do this. You will trash your data, and quickly put it in an inoperable state. Since your using dropbox, reverting back to a operable state will be extremely difficult or impossible, depending on how well you did backups and timed them.
Your going to host the website somewhere, no? Don't they provide a mysql db for you to use? If so, you should check and see if you can expose ports for remote managing it, or just manage it through the hosts phpmySQL interface. Unless your making structure changes, you probably don't need to be in deep with the database anyways. You say that you have the DB on your machine but your friend can't access it, maybe you can open a port for him?
Finally, even if you did get the files to sync without trashing your data, he's still going to have to setup a MySQL instance on his side to do something with the datafiles. MySQL consists of two parts, the software and the data, you can't just dump the data into dropbox (or any sharing method) and have it be available to someone else who has access to the datafiles. If I'm mistaken and he does have mysql server configured, why can't he just install his own set of tables?