There must be a dozen ways to remove rusted, broken, stripped and just generally uncooperative screws, nut and bolts.
I just removed a seized screw from a steel sliding screen door yesterday using a 4-piece kit of screw removal drilling bits. The bits have a v-shaped "blade" that basically drills into the screw head until it's gone. Since I was chopping the door rails to a smaller width, I didn't have to bother removing the screw shaft.
I also have a set of reverse-threaded bits that bite into a screw head when unscrewing it, for those stripped ones.
Most times, if a screw is stripped, I take a Dremel cutting wheel and make a slot as deep as i can in the screw and use a slotted screw driver on it. Works great.
Then there's the vice grip method if you have the angle and room to get a grip on the screw head. Not a good method for counter-sunk screws, though.
Heat is another option, depending on the materials involved. A torch used to heat a steel bolt in a steal part can cause the metals to expand then contract, allowing enough movement to break them free. I wouldn't suggest using a torch on aluminum or other soft material.
WD40 makes some very good rust removal products. There's one super penetrating formula I've had luck with, and the original does a good job, too. You do have to give it time to work, and maybe apply multiple times, but eventually the spray usually penetrates enough rust to loosen it.
One of the best things for removing corrosion is vinegar. I had 2 quarters this week that were wedged in the gap of a window screen for a decade that were unrecognizable. I soaked them in vinegar for a few hours, and 80% of the corrosion was gone. A little polishing with a scrub pad and brush, and both are nice and shiny.
Unfortunately, by trying to defeat the rust, you usually wind up stripping or breaking screws, making it even harder to remove. You then wind up needing a combination of remedies.
If you don't have the time, tools, or patience, sometimes an "upgrade" is the best option. Otherwise, you might spend money on the tools and sprays, only to inadvertently damage the faucet and having to get a replacement anyway.