After Aunt Bee backs into the tree, the car is either on the sidewalk up against the tree, on the street one foot from the tree, or without damage...depending on the scene.
As Aunt Bee and the bystander examine the damage to the car, they are looking on the wrong side of the tree. The damage was between the wheel well and the rear bumper, which is to the left of the tree from Aunt Bee's perspective. But they are looking to the right of the tree, at part of the car that wouldn't have even hit the tree.
Right before Aunt Bee backs into the tree, her hat is untied below her chin. Seconds later, after hitting the tree, the hat is suddenly tied.
In the scene where a Bee crashes her car, a close-up shows the damage at the very back end of the passenger side. But when she in Loel are surveying the damage they're looking at the middle of the car.
At the end, Aunt Bee is taking the family for a drive and instructs them to put on their seatbelts. They all pretend to do so. Seatbelts were not an option on this car in 1955. The first year they were available was '56.
Aunt Bee is taught to drive a manual transmission car (with a clutch and "3 on the tree"). In fact, the car has the "Ford-o-matic" transmission, which is automatic. There's no clutch in the car. The Fordomatic emblem can be clearly seen on the trunk of the car.
When Goober is teaching Aunt Bee to drive it's easy to see that a stunt driver is used in place of her.
In the driving scenes, Aunt Bee is not actually driving the car. It is a man...dressed in a hat and dress.