I believe you need to read up on classes and object here.
Apache is a forked daemon.
When a page request is made, Apache forks a child that is your apache user session.
All PHP, HTML and other such noise as DB connections are completely abstracted, as they only exist in your child process.
So, no... there will be no overlap here.
The user will get their own connection via their own instance of the DB_SQL wrapper.
The only time you need to be worried about user access conflicts is with unique file resources, that is actual directories and static files on the filesystem, which is why there are tools to flock and such.
Also, a database is a file just as any text file. So if you are having large executions take place, it would be recommended that you either LOCK the table while writing or degrade the importance of the query so it executes out of order to ensure that normal users have priority.
But, considerations such as these are for queries that are ALTERING 100's of fields of data or more simultaniously and would take more than say 1ms to complete (your standard query should be say .25msecs).
hope that helps you here.