^Plastic sheet keeps moisture away from the car's underside, possibly preventing any undue corrosion. I park on plastic too. Buy at Home Depot- painter tarp 3-4 mil.
Yes, use a fuel stabilizer, like Sta-Bil. Run-in the product a few minutes, prior to shut down- so it gets into the entire fuel system. Make the tank as full-full as possible, as this keeps out air and potential condensation/moisture.
I move my car a few inches, at the beginning of each month. One month forward, next month back. Then I do this weekely, the last month of hibernation. Very easy to do. Necessary.....? I don't have any flat spots, so I'll continue to do so. Some tire brands are VERY prone to FS. Mine, are Michellin, and not known to have this issue. But, like I said easy to do, so I'll continue.