Does a PHP exception stop execution?

Yes, uncaught exceptions result in fatal errors that stop the execution of the script. So the do_some_database_stuff function will not be called if an exception is thrown. You can read more about exceptions in this article.

Have a look at the PHP manual on exceptions.

When an exception is thrown, code following the statement will not be executed, and PHP will attempt to find the first matching catch block. If an exception is not caught, a PHP Fatal Error will be issued with an "Uncaught Exception ..." message, unless a handler has been defined with set_exception_handler().

php.net/exceptions

So yes, the rest of the function is not being executed, a fatal error occurs instead.
If you catch the exception, execution of the script continues in the corresponding catch block, everything "between" the function which throws an exception and the catch block is not executed.