I have two issues with my kitchen sink right now and am looking for any suggestions/advice on how to diy?
(1) the faucet is leaking under the sink
(2) hot water valve isn't shutting off the hot water all the way
Is the faucet itself leaking or is the seal between the faucet and the sink faulty? If it's seal, loosen the nuts securing the faucet, clean the surfaces and reapply plumbers putty. Resecure the nuts and all should be fine. If it's th valve that's leaking, it will depend on where the leak is whether rebuilding the faucet will cure the problem or if replacement is required.
The hot water anglestop under the sink sounds like it's faulty and should be replaced, however due to age and corrosion that may not be an easy task. A suitable alternative would be to turn off the water main where it enters the house OR to shut off the water flow at the hot water heater.