"I put a heavy book on top of the crumpled (wrinkled) paper to flatten it (out)" is the construction I would use in the first sentence. I'm assuming in this context that you mean that the piece of paper is wrinkled rather than a crumpled up wad of paper.
"My cake got squashed in the bag" is what I would use for the second example, although, crushed would not be technically wrong.
Crush seems to be used a lot more with objects that are hard or idiomatically and squash tends to be used more with objects that could be considered "squishy" in my opinion.
