#1 Yes you can start the generator while connected to shore power; the transfer switch will transfer power to your unit as Jim@HiTek noted. I normally disconnect the shore power before starting the generator just to make sure the transfer switch operates properly; maybe I am just being overly cautious.
#2 Exactly as Jim@HiTek stated. We run our front AC at night and close most of the front area air outlets; this in turn pushes the conditioned air to the rear bedroom area. It keeps the bedroom very cool without the noise of the back unit running.
#3 As I stated in #1, I would definitely unplug from shore power. After a power outage, when power is restored, you may get a high voltage spike that could damage your electrical system and some of your items such as microwave, televisions, computers, etc.
Ray