How many puppets.

Using the definition given earlier about puppetry: the purposeful and thoughtful animation of an object, there would be five. You may disagree with as many as you wish.

Paddle Ball: Single string marionette.
Balloon Dogs: Overt table-top puppetry.
Kleenex: Overt table-top puppetry.
Pong: Shadow puppetry.
Dr. Storm's Lament: Object or icon puppetry.

Not puppetry
The opening of the show. None of the suitcases are animated. They are just moved from one place to another.
When Gil Travers takes tie off of his teddy bear. He simply playing with it.