What parts do I have to replace--and how to remove them--when the threads that hold the retaining nut of the pivot rod (which operates the sink stopper) have deteriorated and consequently leak? Is there a fix that doesn't entail detaching the pipe from the sink, or must I have an entirely new pipe installed. I'm guessing that Moen would specify the precise piece. Lots of hassle for one small failure. What would have caused the threads to deteriorate in less than 20 years?

Its call a sink tail pipe (with pop up). Very inexpensive and easily replaced. Disconnect the drain trap. Disconnect and remove the lift rod. Loosen the mounting nut (up next to the sink). Then hold the flange and unscrew the tail pipe from the flange. Only difficulty you could have is holding the flange while unscrewing the tail pipe. But since you will be discarding it, you will not be concerned with scratching it. If necessary hold it with a pair of pliers.

Use plumbers putty between the flange and the sink when installing the new one.

It is not Moen specific. If you have a Moen it is only because it came as part of a Moen faucet package.

I agree with Hkstroud see the image below this what he is talking about, good luck.

John