Most electronic failures end up being in connections. If you have a satellite design and go to the reliability guys, what they want to know is how many connections are in the bird and what kind. The fact that this is temperature-sensitive also points to connections. That's why, as Piloto suggested, disengaging and reseating connectors is always hign on the troubleshooting list.
When you attempt to transmit, is it that ATC cannot hear you or that you cannot hear yourself on a hand-held in the airplane? If you can hear yourself, then the radio is probably transmitting just fine and it is an antenna connection issue. Deteriorated coax usually just adds attenuation, it does not get intemittent.
If you can't hear yourself locally, then it's in the radio as you expect. The fact that the nav flag moves (something almost totally independent of the primary problem) tends to point toward a grounding problem. A weak ground connection, inside or outside the box, can produce very weird behavior. Can your avionics shop leave the box in a fridge overnight and bench test it when it's cold? (I suppose you could do this test if you can get the radio thoroughly chilled and quickly plug it into a warm airplane.) That might provide a clue as to whether the issue is inside or outside the box. If they can duplicate the problem, then it becomes Garmin's problem not yours. If not, you still own it. Troubleshooting this kind of thing can be a b1tc#.