a) Are you sure you are using the new Security+ system? I am about to buy a system and don't want to be having problems with new neighbours or door security.
The new rolling code technology (Security +) feature is offered on the Craftsman and Chamberlain Garage Door packages. Likely Craftsman copied or bought trademark and logo replacement rights to remarket the Chamberlain package under the Sears "Craftsman" name.
I suspect that it is possible during a communication disruption to lose synchronization between matching xmitters and receivers. This could lead to your unit opening the neighbour's door.
The key differences seems to be that the keyless or keypad opener is wireless for the Chamberlain while it remains wired for the Craftsman unit.
My suspicion is that the wired unit handles colder weather better (wireless depends upon batteries which lose capacity in colder weather). It is also likely that the wired in units bypass the rolling code technology altogether.
This creates a security breach as one can simply rip the keyless entry from the side of the building and short the remaining wires to open the garage door.