Yeah, it seems like John usually takes care of mundane sorts of things like buying milk, so driving seems like a natural extension of that. It seems like a task Sherlock would deem boring unless it took the form of a high speed pursuit. Then again, Sherlock does love leading John around, and I guess that would make him inclined to take the wheel. I do like your idea of them bickering like Mulder and Scully.

Were there jokes about John's little feet not being able to reach the peddles?

Well, it was a rather large car. ;)