If you answer anything as a "yes," the affected individual will be seen by one of the nurses who works on the ship. If the nurse says you can't board, you do have the right to request to be seen by the doctor (but don't expect a different verdict). If they say you cannot board, you are out of luck. THey do not quarantine you on the ship; they deny boarding. They used to send you to WDW, but no more. Now you are just out of luck and hope you have travel insurance. It never did make sense to me that people were supposedly too ill to cruise, but healthy enough to go to WDW and expose other people to their illness.
Yes, you can go to a local urgent care center and get a note from whomever, but the ship's medical staff has the final word. They do not have to accept whatever an urgent care center, ER, or your home doc has said. That said, if you have a chronic medical condition that causes coughing, runny nose, vomiting, or other issues, it does help to have a note from your physician documenting this. For instance, if you vomit or cough due to reflux, that note will probably influence the staff to allow you to cruise since you have documented the situation.
Careful hand washing, stay healthy, and get on board!