what is the place called where cars are sold

It depends if the cars are new: a car dealer, if they're second-hand, it could be a garage.

In the U.S. a 'car shop' would be a place where cars are fixed, e.g. I am taking my car to the shop - I am taking it to be repaired.

Cars, both new and used, are sold at a "car dealership". A small establishment which sells only used cars is sometimes called a "used car lot".

a dealership

dealership, dealers (-s on the end same as in butchers, bakers - it's debatable whether there should be an apostrophe with the s but you commonly see it without).
Also car showroom(s) is in quite common usage here in the UK.
You might also hear garage used as a general term, though that probably implies that repairs as well as sales are carried out there.

In the U.S. a 'car shop' would be a place where cars are fixed, e.g. I am taking my car to the shop - I am taking it to be repaired.

I don't know about that...I agree about the car being in the shop, but I have never heard the expression "car shop."

Also, garage is not used in the US for a used car dealership. A garage (or "the shop") is where a car is repaired.

Here (NZ) we say, car dealer, or car showroom for new cars. For used cars, car yard.

Garage or workshop for repairs. Panel beater for repairing dents.

Yep - as above - it NZ i would call it a car dealer or a car yard.

Or, a cheap way to get a car to drop someone off at the airport and not pay for a taxi....

RE #5,
I would not say Car Shop, but just The Shop to refer to the generic place I go to get my car fixed that is not The Dealer. If I am feeling rich and take my car back to the place that I bought it to get serviced, then I would say I'm getting my car serviced or fixed at the dealer. At this point everyone around you will moan and tell you about their favorite shop where you should take your car instead.

"My car is in the shop" would mean I'm having work done on it, and "the shop" I'm referring to could be called a "body shop." There are now gyms with the cutesy name "Body Shop," but that name has been taken from its earlier use to mean a place that did car repairs.

"Body shop" itself was used because some work done was not just a tune-ups or oil change, but work on the car body itself, such as the repair of dents, replacement of a damaged door, etc.