All three of your phrases are grammatically correct and their meaning is crystal clear.
However, none of them are what one would expect to hear in everyday colloquial English. As a native speaker I would say "I don't have a car" or "I don't have any cars".
In my efforts to learn languages other than English, questions like this bedevil me constantly. It is so easy to say something grammatically correct that everyone will understand, but that will immediately mark one as a foreigner. On the other hand, it is very hard to learn which are the most commonly used phrases, the things that mark one as a native or truly fluent speaker.
The "I have not a" expression, while probably correct gramatically as Probus suggests, is simply not used. We use instead, "I don't have a" or - perhaps more poetically, "I have no". I could say, "I have no reason to do this," or "I have no car to drive there." - you'll see in these examples that there is a phrase following your admission of non-ownership. While it would be correct to answer, "Why did you do this?" "I have no reason" - the more common or natural expression is, as Probus suggests, "I don't have a reason" or "I don't have a car."