When Wally and Ward are test driving the convertible, the rear view mirror is missing; however, the very next scene, when they return to the car dealer's lot, the mirror is now mounted on the car where it should be. Then when Wally and Ward get home after buying the car, the mirror is still on the car; however, when Wally, Ward, June and the Beaver take Wally's newly acquired car out to dinner, the rear view mirror is again missing.
Before Wally gives Frank a down payment of five dollars for the "bomb," the car seems to run quietly and without smoke pouring out the exhaust pipe; however, when Ward looks at the car, then these problems of noise and smoke appear.
One of Wally's pros for purchasing a car is that he could drive Beaver to school and save on bus fare and he could also drive Beaver's classmates to school and charge for the service. It has been implied that Beaver walks to and from school. If he took the bus, it would more than likely be one issued by the school like he did before.
As Wally backs his car out of the driveway when he and the family are leaving to go to dinner, there's a couple walking down the street. The man casually puts his right hand in his pocket. The camera cuts to a closer shot of the couple and now, it's his left hand in his pocket.
After Wally drives his new used car home to show it to June and Beaver, Beaver opens the car's trunk and surprises Wally by mentioning that there are two spare tires in there. It is highly unlikely that Wally and his father wouldn't have already looked inside the trunk just out of sheer curiosity, if for no other reason, not to mention making sure there is a spare tire, jack, etc. and no holes or structural problems. Ward would have been too smart a buyer to have overlooked something like that.
As Ward and Wally stand by Wally's new car talking, Ward says that "sometimes a father's standards are even higher than those of the Department of Motor Vehicles." In most states, the agency that issues licenses is indeed called the Department of Motor Vehicles (DMV for short), including California where the show was filmed, though not set. But in Wally's License (1962), the Cleavers' licensing agency was called the Motor Vehicles Department. Calling it the DMV was probably a subconscious slip-up on the part of the California-based writer who wrote Ward's line.
Wally drives his newly acquired Chevy home with Ward as his passenger to end a day of car shopping. Actually, two vehicles should have been driven home by Ward and Wally respectively, Ward's Plymouth and Wally's Chevy, since it was Ward who drove Wally and himself around to do the car shopping in the first place, first to look at the "bomb" and then to go to car dealerships.