micky wrote

Wanted to check tire pressure and couldn't get valve cap off. Got pliers
and it turned but never came off. Noticed the whole stem was rotating.
(Had probably gone once or twice around by the time I noticed it.)

Is this permitted? Tire is not going flat, but should I get a new valve
anyhow?

Yes, if only because it will then be possible to check the tire pressure
again.

Or fill the place where it meets the rim with glue? What glue?

No harm in trying superglue.

I got the cap off with a sharp knife and added a
little air, which has not gotten out yet. (Can a
metal cap get stuck on like this plastic one did?

Yes, they tend to corrode more.

That would be a lot harder to cut off.)

Yes.