The derailleur will move just as far as the shifter tells it to. But the shifting system *is* a system, and it all works together. If your cable outer housing has a kink in it, it makes it harder to shift. If your chain is worn or your cogset is worn, that can lead to skipping. If your rear hub is out of adjustment, that can lead to "ghost shifting." If your shifters are getting worn, that can be a problem too. New cables are a way to eliminate one variable, but be sure to get 'shift' housings and not 'brake' housings. Shift housing is designed to not compress too much over time, which brake housing is known to do. So turn the bike upside down and look at the rear derailleur. The pulley closest to the cogs, the 'jockey' wheel (the other is the 'idler') should be directly in line with the cog above it. If it isn't, adjust the cable tension until it is. If your hanger is bent, it should be immediately apparent; tweak it the way it needs to go with a Crescent wrench closed tightly around it. There's nothing terr...
To add to what I said upthread, the "end"-- for that's what it's actually called, either Mountain or Road-- is actually a zinc alloy that is hot-poured and cast in place around the end of the cable. I've toyed with the idea of making my own (there are some odd, sui generis French brakes that take a specialized straddle wire) but at the end of the day, it's really a false economy.
If your rear wheel has a freehub (as opposed to a freewheel) Surly makes a single-cog cassette that just slides on and locks down like a gear cluster. If you have a freewheel, there are single-speed freewheels available from Shimano and Dicta (to name two brands). You'll need a freewheel tool, a vise and possibly a chainwhip to get the old one off; bear this in mind as opposed to simply hiring it done by the bike shop. If you really want to do it right, you'll re-dish the rear wheel so it's more nearly symmetrical, and that's when the fun really starts. There's something very Zen about wheelwork. There are loads of tutorials available online; I particularly like the ones available at Instructables.com
As much as I loathe the idea of replacing the brake cable because the swaged end is mangled, I think it's the right thing to do (and I work in a bike shop that specializes in used bicycles and vintage repair, I'm not one of the new breed who can;t overhaul a 3-speed hub). It's a very inexpensive part, five bucks tops; and the old brake cable is of course re-purposeable for any number of things. There is nothing on a bicycle that's complicated. Almost every single part can be overhauled or rebuilt.