Is there an electronics issue with the opener, or, by chance did someone else purchase a new garage door opener in our neighborhood that uses the same frequency as our garage door?

If one of your neighbors remote controls is inadvertently programmed into your receiver, you door will open when their remote is activated. If this happens you will need to clear out your receiver and reprogram your remote controls. Press and hold the smart/learn button for 6 seconds or until you see the small LED light near the button turn itself off. If the above steps have been followed and the symptoms continue, there may be an electronic short in either the wall control button or the receiver logic board. In this case, remove the wall control wires from terminals 1 and 2 on the motor unit. Should the symptoms persist after the wires are removed, replace the receiver logic board. If the symptoms stop with the wall control wires removed from the motor unit, replace the wall control button, and the wires leading to the wall control button.

The remotes are gray with blue buttons. I know the large button which extends all the way to the side of the remote case is to open and close the door, but what do the 2 smaller buttons do?

I have zero confidence in this opener, so I leave it unplugged all the time unless it's actually opening or closing the door.