I had a problem with my garage door opener doing a phantom opening a while back.
I noticed that it correlated when I went on the air with my Ham radio. Whenever I transmitted on 20 meters (14.3MHz or so), the door would open. I don't know about this interference closing the door, because I never left the door open when I was on the radio.
The RF was being picked up on the wall push-button wire. The tech support was very good for the opener company and they recommended a few turns of the button cable through a ferrite filter. It solved my problem quite well and it didn't happen again.
However, I doubt that the <8mW signal from the Davis would be causing this. I had a 200+ Watt linear amplifier on my Ham rig. Also, the Davis frequency range is in the 902-928MHz band (33cm)
It might not hurt to try temporarily disconnecting the pushbutton wiring at the opener for a while to see what happens. You could also try wrapping the opener pushbutton wiring around a ferrite. Again, this would be located close to the opener.
Greg H.