Pete, It is so funny (now) that you are writing about this problem as we just went through the same identical problem. We did all of the things you did and the problem still existed. We even thought that maybe someone in the neighbor maybe had a new electronic toy that was causing the problem (don't rule out that possibility). Anyway, what we found is that the garage door opener has a little strip of metal inside of it and when you press (or click) the opener, it simply causes a short circuit that makes the door go either up or down. Over time, the piece of metal lays kind of close to the contact, like really really close and simple changes in the temperature can cause this strip of metal to move sllightly "short out" thus causing the door to phantomly go either up or down. If the opener is on the visor, then just getting in and out of the vehicle can cause a movement that again will cause the short circuit. Caution, when we took ours apart and finally found the problem, I decided to just bend the metal back away from the contact. The opener was so old and the metal had been bent over years that it broke off.
Lastly, (after all of that), a weak battery in the opener can cause phantom opening and closing.
Hope this helps and sorry to be so wordy. Terry Alexander