Discussions of secure PHP coding. Security in software is important, so don't be afraid to ask. And when answering: be anal. Nitpick. No security vulnerability is too small.
Yes, as long as the MySQL server allows connections from either all hosts or the specific host PHP is on, and the MySQL user you connect with is able to connect from the same host.
Do you have administrator privileges on the MySQL server?
Alright, then it should be easy enough to give the user the permissions they need to access the database(s) from that address. Don't forget to make sure your MySQL server's port 3306 is not blocked, and make sure that it uses TCP to listen on 3306 (take a look at the skip-networking config option, make sure it's commented out in either the server startup or the my.cnf).
Also, make sure the MySQL user is given just enough permissions to do what it needs, nothing more. That probably means restricted to a few tables, and ability to select, insert, possibly update, but nothing else.