The base is likely brass, so it won't rust. It's just a cosmetic issue. You can try touching it up with a black sharpie marker.
You need the correct size wrench or a large enough adjustable wrench. Run by Harbor Freight over in Salinas and pick one up for a few bucks. Or check with your neighbors. Someone is bound to have a wrench big enough to loosen it up.
I usually unscrew my antennas each time I wash my truck. Partially to wash underneath, partially to check for corrosion or any other issue.
Brass corrosion can, generally, be broken with a bit of shock loading. (Part of why firefighting appliances are made of brass.)
As mentioned, get a wrench that fits. If it is beyond hand tight, with a properly fitting wrench, some light tapping on the wrench handle should break it loose. (If not, apply some PB Blaster to the threads, let it wick for a few hours, and try again.)
I also remove my antennas when washing vehicles. Once you get the antenna off, you may want to consider using a bit of silicone dielectric grease on the base threads. You don't need much, just enough to lightly coat the threads. This should make the antenna easier to remove in the future. As far as the scratch goes, it's mainly a cosmetic issue but a Sharpie or some touch up paint should fix it up just fine. (Be sure to apply paint with it off the vehicle though. I don't want you cursing me if the paints gets spilled on your car ;-)
Too much torque on the frozen nut can distort the sheet metal. Some penetrating oil will help. Be careful of the paint. Careful tapping the nut on all sides will help loosen it up and wick up the oil. If it still does not budge, heat up the nut with a hair dryer all around and it will expand a bit.