Mysql functions for web apps
I can't answer to question 1 as I do not have enough information about the MySQL instances offered by different hosting providers.

To answer the rest of your questions, yes, if you can leverage the features of the RDBMS you're using, you might replace PHP code with SQL scripts that can run faster than your code. Just think about the time you're saving up by not transferring your data back and forth.

On the other hand you would probably still end up using PHP code or other programs (crontab or any other kind of scheduler) to run your timed tasks as there's no built-in scheduler in MySQL.