"I assume all of this is true of the City tag too?"
Yes, though depending on where you look there may be also census boundary tagged as admin boundary, or no city boundary at all.

Smaller areas only very rarely have a boundary relation mapped, depending on their (legal) status.

"Do you know of a way to extract the county administrative boundary for a node?"
No, I'm not a developer, so I don't have experience with this. You should use a geocoder like Nominatim, or related application.
https://wiki.openstreetmap.org/wiki/Nominatim
If you need help, you can ask on the forums
https://community.openstreetmap.org/c/help-and-support/7/none