Is it possible to make half a car outta foam??

It would be very expensive and time consuming if your after a realistic look and not a cartoon appearance. If its indeed for a music video, id suggest doing it CGI or actually cutting a cheap car in half. Will be much cheaper and easier.
expensive and very time consuming, unless you need something very specific then it may be cheaper to rent an exotic car for a day.

Here's a neat build thread about a guy who made a plug mold of a Bentley Speed 8

Lots of chassis and engine stuff to dig through, but the body build is towards the end.
Would it be hard? Yes, as difficult as building a car. It only looks easy in the pictures. You don't know how much work (+$$) went into each photo.

I agree with the opinions above, rent a car for the day, go to a junk yard and find a car smashed one side and shoot the other side, OR: Build a model car and do some green screen trickery in the computer.
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