Hello,
I just replaced my faucet and I had a really hard time getting the shut off valves to shut off. So, I replaced the innards of the "newer" valve (circa 1980). The problem I'm having is the older valve is a different type(circa 1960). Can it be unscrewed (forced) off the pipe and replaced with a new valve or must I cut and thread the pipe to do so? See attached, thanks.

If you have a male iron pipe thread that the old cutoff valve is screwed onto you can unscrew the old cutoff valve and use some teflon tape on the threads and screw the new valve on. I would advise you to use a back up wrench on the pipe as you unscrew the old cutoff.

If you have copper coming out of the wall then that is another story. So let me know what you have.

You do not have a threaded angle stop. What you have is a compression type valve. This can be replaced simply by unscrewing the compression nut from the valve body, removing the old angle stop and allowing the old compression nut and ferrule to remain on the pipe and installing a new compression angle stop using the old nut and ferrule. Good luck, Tom