To be able to truly short something you technically need to be able to borrow the security so you can sell it. There needs to be a system for borrowing in place to be able to do this which is very robust for large U.S. stocks but doesn't exist for CDOs mainly due to the complex legal structures around them.
However, the word "short" is commonly used in finance to mean profiting from a loss of value of something. So the use in the movie title, though a bit confusing, was fine.
Credit Default Swaps are not technically insurance as you don't need own the thing you are "insuring" (and for a few other reasons). However, I agree with the movie that thinking of them as insurance is a pretty good way to understand them. As you are playing a monthly premium to for a contract that pays out when something goes very wrong.
However, the movie was a little fast a loose as CDS of various types were regularly traded well before the bubble even started. Though maybe not that particular type of CDS. So while they may have been "expensive" it was the easiest route and reasonable idea.