Is it possible to vomit poop?
It is not possible unless the person is suffering from some extremely rare condition or disorder. I had a roommate once whose summer job was to administer barium enemas to patients in a hospital. She told me that one patient vomited the enema.

Is it possible for a man to have poop come out of his penis, or for a woman to poop out of her vaginal opening?

Not normally; fortunately the plumbing of the genitalia is entirely separate from the plumbing of the digestive system! However, there are certain pathological situations that can cause the pipes to get connected together wrong. Cancer can do it, as can surgical diversions of the human tubing. Such fistulas, as they are called, can cause feces to come out of the urinary system, or urine to come out of the anus.