If nothing shows in the logs, you might check your wiring - even outside of garadget. If you have bare wire shorting (touching) it would act as a button push. I've had this happen, even pre-home automation days. Admittedly that was with an older opener so I don't know if things have changed to eliminate that issue.
As to the false alerts - I had that issue on windy days but I purchased some extra reflective tape that cut down on the issue drastically. The other thing I've done is have a camera set up in the garage so if I get an alert I can just check the camera feed to see if it is legit or not.
Finally, my garage door opener has a vacation mode that locks it out from opening (yes, I could just unplug it as well.) I have to go through the front door when I return, but that's not usually an issue as I'm generally taking a taxi etc home. You might consider that for some extra peace of mind.
Hope some of these thoughts help, but like suggested I'd say try to fix the false alerts stuff so you can feel more comfortable.