Does a power outage / unplugging damage major Appliances
It certainly doesn't help. Dropping power off and restarting of any piece of equipment without going through a staged shutdown or startup leaves the equipment in a state where the normal sequence is interrupted and leaves the controls "confused" so to speak. Also every new appliance you buy generally has a microprocessor and electronic controls these days and they are a little more sensitive to surges and low voltage outages. During an outage you should shut off any equipment you can that was on before the outage and restart after your are sure the power is going to stay on for a while. What's more damaging to these appliances than a long term outage is the rapid blinking and flashing where say a tree branch is touching a power line during a windstorm. 

All that said most appliances are fairly rugged and can take a certain amount of abuse. 

I always suggest that a whole house surge suppressor be install at your main panelboard. That also helps cut down spike that may enter your system.