A cut bill is perfectly valid at least in Europe and I suspect in the US is the same as there are movies in which the Big gangster cuts a money bill and tells someone else if he wants the other part do something.
Yes. If you have both halves of the bill a bank will exchange it for another one. Even if you're missing a part of it. US bills have two serial numbers; generally you need one whole serial number and part of the second. (Obviously so you can't cut it in half and try to exchange each half separately.)
The fake money already has a bunch of features that make it clear that it's fake if you actually look at it. Cutting it in half could work, because it'd break the routine of the person accepting it and make them pay attention.
