how does deskzilla work: does it use http requests to bugzilla server or it connects to bugzilla database?
so, do I need to open access to mysql server, which holds my bugzilla, to external connections?

Deskzilla uses HTTP to access publicly available Bugzilla pages, just as a human user does with a browser. So you don't need to do anything extra on the server side to use Deskzilla.

Best regards,
Igor