Plumbing – What brand of faucet is this and how can I hook the sink sprayer up to it

I would just replace the hose. You would probably have to take it apart to clean it anyway. I found replacement hoses for around $4. The directions seem fairly straightforward.

Sounds like the packing inside of the faucet needs to be replaced, assuming this is similar to an outdoor hose faucet. After you shutoff the water (probably for the whole house if you don't have a separate shutoff for that line) you can open the valve up by unscrewing the nut that the valve stem goes through. You can find new packing material at most home improvement stores that looks like a black string like material, typically right next to all the washers. I'd also suggest replacing the washer at the same time. And clean all the threads and apply fresh plumbers dope before screwing things back together.

If the installing plumber was nice, you can also unscrew the entire faucet from the pipe. Then clean up the threads, put on some fresh pipe dope, and screw on a new faucet. It will cost a few bucks more, but the process is much easier and the result is also nicer. Unfortunately for me, my plumber decided to solder the threads making this task nearly impossible without opening up the wall to cut the pipes.

If you're not comfortable with soldering pipes should things go wrong, and you have to shutoff the water to the entire house, then consider having an appointment with a plumber already scheduled, that you can cancel without incurring any fees, before starting.