Presumably your minis go through some sort of amplifier and/or speaker systems for their sound. Route the cables through a mute switch next to your PA system. Otherwise you'll have to go to each mini and mute them individually.
As for pausing, that's application-specific. You can get remote network controls for iTunes and VLC, but unless you know what application is doing what, you're out of luck on that front. You may be able to write an Applescript that can be triggered remotely to stop *everything* but there's nothing (that I know of) that works "out of the box".
Or, you could just invest in a louder PA system. 😉