When one retrieves a database $link with mysql_connect(), how can one check if that link is still valid after some time? (I am not asking whether the returned is valid, I mean later)
You can use mysql_ping() or mysqli_ping() for this. However, in a regular web application you should just execute your queries and abort with an error if a query failed (or retry once).
There's no reason to test.
Just execute the query, and you will see...