turning my system on during lightning storms

All the above is great advice. Also, the satellite dishes can carry spikes and lightning into your receiver, as can FM antennae. I am having an electrician install ground wires and rods for my dishes and antennae, and I unplug the things I don't want destroyed. Even small static discharges can harm delicate electronics.

Grounding antena won't protect from direct hit (even ground rod is problematic). The real purpose for grounding antena is to prevent static electricity build-up caused by the wind that can charge your antena to several kV making it electrically the "tallest" structure in the area. Size of the grounding wire does not matter.

Always unplug to prevent ground differentials.

You must live in the midwest. The other night a particularly lively storm came up fast. I jumped out of my chair to pull the plugs. Definitely the safest bet.

I wouldn't chance it. As soon as I hear thunder I run to the system and unplug all of the components from the wall. Unless your system is a GPX boombox, I'd unplug.