A folding chair folds into a folded chair.

A folding chair is 'une chaise pliante' in French.

You would buy a folding chair if you need to have a chair that can be put away in a corner to save space. By being able to be folded, the chair does not take up as much room as a regular non-folding chair.

You likely heard it wrong. A folding camping chair would be like any other folding chair, but designed to stand up to camping conditions. A folding campaign chair likely would not exist because one rarely has idle time when on campaign.

Yes, you can find folding chairs that have arms and are not the lawn chair kind. You can find them at the folding chair shop upholstered, plastic, and wooden.