This sentence is ungrammatical:
Did you clean the car and drove it?
Since did governs both the verbs, clean and drive, these verbs must be in their infinitive form.
This sentence is grammatical:
Did you clean the car and drive it?
The word you is omitted before drive, since it's the subject of both verbs governed by did and it's already been stated. You are correct that this sentence means the same as:
Did you clean the car and did you drive it?
Very strictly, this means the same as asking two independent questions:
Did you clean the car? Did you drive the car?
The answer would be yes if the listener did both of those things.
However, normally actions in a sequence connected by and suggest a chronological sequence if such a sequence makes sense. So, the way someone would normally understand this sentence (unless something in the context prevented it) is:
Did you (clean the car and then drive the car)?
where the listener must have done the part in parentheses in sequence for the answer to be yes.
Someone might answer your original sentence (b) like this:
No, I drove the car and then cleaned it.
Notice that someone would add the word then to clarify that they're talking about the sequence of events, not just whether both events happened.