The fact that the headrest can't be lowered indicates that broken retaining piece I metioned in one of my threads.
As far as the unsolicited raising of the headrests...you need to find an exorcist!!!
Seriously, that sounds like a shorted switch or wiring harness near the headrest. Does pressure on the seatback or cushion start the headrest moving? If it does, the culprit is wiring at the headrest mechanism. Otherwise, it's a switch problem.