How can I remove a faucet nut that is completely frozen with corrosion?

That sounds like a classic mini-widespread setup where the removal process is acutally the opposite of what you'd expect. Instead of just fighting those bottom nuts untill the sink cracks, you ussually have to pull the handles and decorative escutcheons off from the top first. Once those are out of the way, the valve bodies shold actually drop down through the holes rather than pulling up. It’s a huge pain when they're seized, but going at it from the top is usually the only way to win. If the mineral build-up is that bad, PB Blaster is a start, but you might need to scrape away the heavy green crust with a flathead screwdriver first so the oil can actually reach the threads. Those old plastic supply lines are prolly brittle anyway, so don't be afraid to just cut them out of your way with some snips. It clears up the workspace and you're going to want fresh lines for the new faucet regardless. Check for a hidden set screw under the handle or the hot and cold indicators. Pop those caps off and see if there's a screw holding the handle on. Once the handle is off, there's often a large decorative nut or a flange that holds everything to the sink from the top side. If you can get a pipe wrench or even some large channel locks on that top piece while someone holds the nut underneath with a basin wrench, you can usually break the seal. If the hardware is truly welded by corrosion and won't budge even with a basin wrench, don't kill your back or the sink. I’ve had to use an **oscillating tool** with a metal-cutting blade more times than I’d like to admit. Just be extremely careful not to nick the porcelain or stainless steel. Cutting the mounting nut from the bottom or just lopping the top off is often faster than fighting forty years of calcium deposits. Just make sure you have someone up top holding the faucet so it doesn't fall and damage anything when the metal finally snaps.

I'm literally just starting to learn DIY plumbing and that oscillating tool suggestion sounds terrifying but also kind of genius. My dad always told me to try a hair dryer or a heat gun to expand the metal when things are seized up. Does that actualy help with the green mineral stuff or does it just make everythign too hot to touch? Also, for a newbie, how do you stop the blade from jumping and scratching the porcelain? I feel like I'd slip and ruin the whole sink in five seconds.

Heat guns are hit or miss on that green crust. Mostly just smells bad and doesn't do much for calcium. If you're worried about the blade jumping, try a Dremel with a small cutting wheel instead of the big oscillating tool. It’s way more precise in those tight spots behind the basin.

Another trick is to drill a couple of small holes vertically thru the side of the mounting nut. Once you weaken the metal enough, you can usually just pop it with a flathead and a hammer. It beats fighting the threads for two hours. Just watch your eyes for flying metal bits.