"What kind of car" vs "What kind of a car" vs "What kind of cars"
Which of the following sentences is correct? Why?

What kind of car do you have? What kind of car is that?
What kind of a car do you have? What kind of a car is that?
What kind of cars do you have? What kind of cars is that?

Surprisingly, I've not found a grammar guide where this issue is explained.

'The word "that" refers to a thought or situation, and cannot refer to the plural of a countable noun like "cars." ' said Peter. Interestingly, I didn't mean "cars" by "that", and referred to a single car in all cases. Now I see it where I need to pay more attention. And wouldn't it better to ask: "What kinds of cars are those?" in lieu of "What kind of cars are those?"

Question

"What kind of car" vs "What kind of a car" vs "What kind of cars"
Which of the following sentences is correct? Why?

What kind of car do you have? What kind of car is that?
What kind of a car do you have? What kind of a car is that?
What kind of cars do you have? What kind of cars is that?

Surprisingly, I've not found a grammar guide where this issue is explained.

'The word "that" refers to a thought or situation, and cannot refer to the plural of a countable noun like "cars." ' said Peter.  Interestingly, I didn't mean "cars" by "that", and referred to a single car in all cases. Now I see it where I need to pay more attention. And wouldn't it better to ask: "What kinds of cars are those?" in lieu of "What kind of cars are those?"

Answer

The first and third sentence are in my opinion correct. (The first can be used when you assume they have only one car and the third assuming they have more than one). But I can't really pinpoint to a grammatical rule why the second one is wrong (it's just is), perhaps others can help you with that.

Edit: Just noticed that the third sentence, second part: " What kind of cars is that?" needs to be "What kind of cars are those"? (since it's in a plural form)

Answer

"What kind of car do you have? What kind of car is that?"
- Both of those are perfect.
"What kind of a car do you have? What kind of a car is that?"
- These are not technically correct, but are used in informal speech, especially the second one, "What kind of a  is that?" is an somewhat negative question meaning that the thing in question seems to be of bad quality to the person posing the question
"What kind of cars do you have? What kind of cars is that?"
- The first question is fine, but the second should be, "What kind of cars are those?"  The word "that" refers to a thought or situation, and cannot refer to the plural of a countable noun like "cars."

Answer

For the first and second sentence you can use either, but the first one is probably your best choice. What kind of car do you have? What kind of car is that?
For the third sentence thats just saying that the person you're asking has more then one car and you should say "what kind of cars do you have? What kind of cars are they?"