The car is a 1960 AMC Rambler American Deluxe.
The scene with Goober and Gilley arguing in the street was filmed in front of the three houses on Lillian Way, just outside the east gate to Desilu Studios. The three houses were real and one in particular was used for several character's homes during the series, most famously as Thelma Lou's. Because there were only three residential houses on the block, the opposing shots between Goober and Gilly had to be filmed on the same side of the street, at different angles, at different times.
First of two appearances of Larry Hovis who plays Gilly Walker. Hovis went on to star as Sergeant Carter on Hogan's Heroes after playing a supporting role on Gomer Pyle USMC.
Larry Hovis, who appears here as Gilly Walker for the second time, would be starring regularly in Hogan's Heroes (1965), which premiered later that year. The location-set for Stalag 13 would be built just a hundred yards north of Wally's Filling Station on the 40 Acres backlot.
Don Knotts does not appear in the episode, the explanation is that Barney is on vacation, in fact Andy reads a letter that Barney wrote back home at the beginning of the episode.