First, about the short pipe coming out of the back of my kitchen cabinet. I'm guessing if I cut out the back of the cabinet and the drywall I would find a supply pipe coming up from the floor to a 90 degree fitting to the short pipe. I could use the torch to get the short pipe off of the 90 and solder in a new longer pipe, correct?
Well, if you did cut out a big section of the back of the cabinet and the drywall behind it, yes you could do that, but a smarter plan would be to just cut a small hole in the cabinet back where the pipe comes out, and solder a coupling and a longer pipe onto that short pipe. Then, cover the hole in the back of the cabinet where the piping comes out with a "split escutcheon":
That would look a bit more civilized.
You should also inquire about "Sharkbite" fittings, which you simply push over the end of a 1/2 inch copper pipe to make a connection. I've never used them, so I can't comment on them. But, you should certainly investigate whether they would work for you. Also, one of the biggest manufacturers of copper fittings in the US is a company called Nibco. They have new ways of connecting 1/2 inch copper piping, and you might want to snoop around their web site to find out what they have that might work for you.
Second, my original problem was with connecting the braided flexible pipe coming from the shutoff valve to the 3/8" copper pipe going to the faucet. I had to use a 3/8" compression to 1/2" threaded connector. If I was going to go the soldering route in the future, how would things change? I guess I'm wondering what my other options would be going from the shutoff to the 3/8" pipe at the faucet?
What I would do if I were you is...
Once you learn how to solder, and your compression valve starts leaking or looking like it needs replacing, then cut it off and try soldering a coupling and piece of pipe onto what you have coming out of your cabinet. That way you can just extend your existing short pipe. If it's just plain too short to solder a coupling onto, then use a hole saw to cut a 1 1/2 or 2 inch hole in the back of the cabinet to give you more room. If that still isn't enough, you can cut a bigger hole once you've got a plan in place to cover it, and that won't be very hard.
(I'm thinking you might buy a 6 inch hole saw for cutting holes in ceilings for recessed lighting, and then use a clean out cover for a 6 inch diameter gavanized chimney duct to cover that sucker.)
Once you get a copper pipe coming out of the wall, I'd solder an elbow onto it, then a 1/2 inch ball valve, and then you're never going to have a problem getting from a 1/2 inch copper supply pipe into your faucet. If you couldn't do that, then your faucet manufacturer would be in big trouble and more concerned about that situation than you.
I'm not that familiar with the names of the different adapter fittings, nor do I know what kind of connections are on your faucet, but be assured that you're always going to be able to get from a 1/2 inch copper pipe to any faucet in the world.
When you go shopping for a ball valve, make sure it has a packing nut on it so that you can tighten the packing on it. Some manufacturers are making ball valves without packing nuts. Their position on the matter is: "Our valves never leak past the packing, so you'll never need to tighten a packing nut cuz it's never gonna leak, so we don't need packing nuts." To me, that circular logic completely ignores the inevitable possibility that despite the manufacturer's assurances, the ball valve still might leak past the packing. I wouldn't buy any ball valve that didn't have a packing nut to allow me to tighten the packing just in case.