First - the TV. The coaches are wired so the front TV goes off when the ignition is turned on. This is for safety reasons, and here in PA, it's the law (
no monitor is to be visible from the driver's position). Although they don't enforce where GPS is involved.
Second - Shore power. I've always disconnected the shore power prior to starting the generator, or turning on the inverter. I'm not sure it's necessary, but just not taking chances. I'm already on my 3rd transfer switch
Running the engine connected to shore power is O.K. as it does not change where the power comes into the coach.
One point of curiosity - you stated
"when I turn the ignition on to move my slides"...My slides will not move if the ignition is in the on position, or if the driver's seat is not completely forward.