I've removed everything from the frame except for the most important parts like the handlebars, crankset and shifters, all because they won't budge. I can't turn the stem bolt or the crank caps to get at those bolts, and I can't loosen the pedals either. I've penetrating oil on all of it so I'll try again later.

I think the frame will be ok. I poked and scraped at the rust and it all came right off. I find no holes or thin spots, and I poked pretty hard on the suspect spots. I'll know more when I get the rest of the parts off but so far so good.