For the stuck screw, try boiling the stem for a few minutes. If you don't mind the smell, boiling it in white vinegar is even better.
Can you get small locking pliers (such as Vise Grips) on the screw for a very good grip on it? It may help.
If you're so inclined, it's possible to drill the screw out and re-tap the hole.
A real, old-timey hardware will certainly have the packing washer. They probably will have a matching stem, too.
Home Center? Skip the trip. A plumbing wholesaler may or may not have the washer individually. You might have to buy a kit.
Old fashioned packing string with graphite is my favorite, but that's just a personal opinion.
Best of Luck With It! You'll get it fixed!
Paul