When Art was beating Roger in the car park the car behind them had BFGoodrich tires but this model of tire wasn't released until the late '70s or early '80s.
Throughout most of the episode, Vivian has a cast on her right arm. At the end of the episode as the twins are holding hands and looking out at the water on the beach, the cast is gone.
This is because it's showing the "ghost" of the victim, and showing kind of a fantasy/what-might-have-been where her injury didn't happen. Similar to Cold Case-Bad Night where a guy and girl were in a wreck but later his ghost is shown avoiding the accident.
A public place such as a doctor's office would never have had rock and roll or pop music playing in the background in 1965. Too many adults would have found it offensive, and in any case was still regarded as "kids" music for teens. Any music in stores or offices was always "easy listening," lushly orchestrated standards with no vocals, played at very low volume.