I have a really simple (but possibly silly question!). If a person has php installed on their webserver, but no MySQL database I can allow that user external access the MySQL database on the external host and run queries back to their webserver right and output that data back on their site on a webpage?
Also would there be any major problems here in the time the queries take to process?
You can, I have run local webserver pages against my hosted mysql server. You need to allow the mysql port through firewalls/routers etc between the web server and the db server, and you need to set mysql permissions to allow access to the database from your webserver host.
Beyond that if you're running this over the public internet you need to make sure your security settings for the database are solid and you don't expose something else from the db server that a hacker may be able to use.
Also, if this is going over the cloud, you should remember that the protocols used are not meant for WAN stuff and things will fail and fail badly. And they will be slow most other times. You are better off making something WAN friendly (eg--web services) and having your app talk to those.