The grandmother's cat, Pitty Sing, is named after one of the characters in The Mikado. The grandmother worries that Pitty Sing would asphyxiate himself if left alone in the house, so she brings him on the trip secretly, hidden in a basket.
It's Pitty Sing who causes the car accident by leaping onto Bailey's neck when the grandmother accidentally releases him from the basket. We don't know why the cat leaps onto Bailey: the feline could be maniacal or just plain startled. Bailey, who apparently doesn't like Pitty Sing, responds by hurling it into a tree after the accident.
It's likely that the cat who rubs up against The Misfit's leg is Pitty Sing, although we never get a clear indication either way.