Before I get to the sentences and their meaning, first a short note about there're.
It is used as a contraction, but not very commonly. It is perfectly fine to contract are to 're, but for pronunciation reasons this usually only happens after a vowel sound: they're, you're. The r sound at the end of there makes it more difficult to pronounce for most people, so most will prefer to use there are.
Now, for the sentences. There is an obvious difference between these two:
There is no car.
There are no cars.
The first one is singular, the second one is plural. However, because there is a negation, the result is the same: there are zero cars. So as such, the two sentences both mean the same. They are also both grammatical.
Another way to say the same is using any, and the correct way of doing that (again singular and plural are correct!) would be:
There is not any car.
There aren't any cars.
You could say there are not any cars, but the contraction feels more natural.
Now, if we have four ways to say the same thing, which one do we choose?
Actually, the difference between the four options is quiet small, but there is a little bit of difference in how we feel part of the message is stressed.
If we are talking about a single car, say a car that I have reserved at a car rental, but they messed up my reservation, you could have a conversation like this:
customer: Where is my car?
clerk : I'm sorry, I have no reservation for you. There is no car for you.
customer: well, if my reserved car isn't there, give me another one.
clerk : I'm sorry, but there isn't any car available.
customer: You are a car rental company, and yet there are no cars?
clerk : It has been very busy, and indeed, there aren't any cars left here.
Let's have a look at the messages behind the four phrases in this conversation:
There is no car: you expected a specific, singular car to be there, but it is not there.
There isn't any car: not only the specific car you wanted isn't there, there is no other single car for you.
There are no cars: I would expect there to be several cars, but that is not the case.
There aren't any cars: Really, no matter what kind of car you want, there are none!
Although there is a bit of a difference, most speakers would have no problem interchanging at least some of the options in the conversation. In general, adding any will add some stress to the absence.
Best Answer
This is an example of English using the "apostrophe s" to signify possession. Basically:
Adding the apostrophe s to a noun turns that noun into a possessive adjective, and it signifies that the noun it modifies belongs to the noun you used to form the possessive adjective.
If there is a book, and that book belongs to Jane, we can say that it is "Jane's book."
So, your corrected example should not have an apostrophe s in the final sentence.
To read more about apostrophes in English, you can look here.