Saving face is on a continuum. In some cultures it’s more important than others. In some it has more impact on day to day lives. Is some cultures it encourages people to take responsibility and resign to “save face for a company or the state” even when it’s not that person’s fault but rather the fault of the position.

So yeah I think the majority of societies practice this to lesser and greater degrees. Some have elaborate customs around it, in others it’s almost imperceptible.

No, they just do it in slightly different ways than Americans so it's framed as something exotic and weird instead of typical human behaviour.

It is an everybody thing, why else would the phrase enter popular usage.

There is a term in Chinese that means "lose face". I remember my parents saying it to me when I misbehaved. I made them lose face to their friends. Save face emerged as a way to avoid losing face: https://www.phrases.org.uk/meanings/lose-face.html

At it's basic level saving face is pretty close to minimizing embarrassment.