You cannot host a BYOND world via MySQL/PHP, or any *strictly web* server, for that matter. You will need a shell server on which you can install BYOND and run persistent processes.
There is a BYOND web interface, called DMCGI, but it's not what you're looking for.
In fact, there are a number of threads discussing the very topic. Search for your keywords to learn more.
Short answer. Yes, BYOND can communicate with external applications.
ts