I think we can't use Any like this because it's a determiner and a pronoun used to refer to indefinite or unknown quantities or an unlimited entity.
Referring to Any - English Grammar Today - Cambridge Dictionaries Online:
Any as a determiner has two forms: a strong form and a weak form. The forms have different meanings.
- Weak form any: indefinite quantities
- Strong form any meaning ‘it does not matter which’
We don’t use any with this meaning with singular countable nouns:
Have you got any Italian cookery books? (or … an Italian cookery book?)
Not: Have you got any Italian cookery book?
And referring to Some/Any from "A guide to learning English":
- In general, any is used in negative sentences and questions:
I didn't get any nice presents for Christmas this year.
I looked in the cupboard but I couldn't find any biscuits.
I don't need any help.
She's so rude. No wonder she doesn't have any friends.
So in that type of questions the use of a is more appropriate than any because we're dealing here with singular countable nouns.
So the right statements are:
Is there a car in the street?
Are there any cars in the street?