Can't hook up new faucet to shut off valve
That's a flare fitting on your existing valve. You may be able to find an appropriate adapter, but I couldn't dig anything up very easily.

The valve looks to be attached to your supply line with a threaded connection, so it should be fairly simple to replace the whole valve using a new valve with a threaded connection. Shut off house water, unscrew this one, clean the threads and apply new thread sealing tape, screw on the new one.