Does anyone know what the likelihood is of the plastic wall tiles, in my father in law's kitchen having asbestos mastic? Same question about the formica laminate sheeting on his kitchen counter? We think the items were put in around 1955.
We're trying to update in order to sell, but unsure how to handle these items in case they may contain asbestos.

Dealing with materials from the 1950s is a crap shoot. Lots of materials were fortified with asbestos binders to make them stronger. Testing is easy, but since mastic is not easily made dusty, its not worth worrying about unless you plan to sand it.

My guess is this is an old plaster wall. Personally, I would demolish the wall, so I could easily replace any wiring, plumbing and insulation, then rebuild with drywall. Unless you are really into the restoration of plaster or drywall, why would you do anything else?