Let's say I am closing a door, and my palm's skin gets stuck between the doors, and started bleeding. What word can I use to describe this situation?
For me (west coast USA), it's "I jammed my finger in the door". This seems to be common: Wikipedia has an article for "Jammed Finger" (warning: medical images of x-rays and swollen fingers, etc)
Google also attests to "caught", "pinched", "slammed" and "smashed" as being common.
For more casual terms, I would recommend "I caught my finger in the door". Many doctors and professional places call it "smashed" or "jammed". All seem perfectly fine to me.
However, I might interpret "smashed" a bit differently, as it shows extra force made when you accidentally closed it on your palm/finger.
Do note that catching your palm in the door sounds a bit off, I would say "I caught/jammed my hand" instead of palm.