What is a humane, safe and clean way to dispose of a live mouse caught in a glue trap?
Wear a thick glove, the gardening kind is preferred. Flip a Ziplock or similar plastic bag inside out and cover your glove with it. Firmly grab the mouse (and its associated trap) with your gloved, "plasticked" hand and with your free hand, roll the Ziplock back out so you can seal it completely, with the mouse trapped inside.

Lay it on the floor, preferably a driveway or garage. Whack the poor thing with a heavy magazine or a shoe; make sure you hit its head for near-instantaneous death. It has already suffered enough; this will help it pass on quicker.

After you do this, please make sure you remove all the glue traps your pest control folks installed.