Our question is when does the flight from LAX to Tahiti leave LAX if the cruise itinary says the cruise starts on April 16th? Does the flight leave LAX that same day? On the return trip, the cruise ends on April 26th. Is that the day our flight would leave Tahiti (probably late at night), returning us to LAX on April 27th morning?
First, you are not crossing the international date line.
So, IF you are flying on Air France (as I did) from LAX to Papeete and your cruise BEGINS on April 16th, your flight would begin at LAX on April 16th at 12:20 AM (just after midnight) and arrive in Papeete at 5:30 AM (same day). Your cruise ends on April 26th - the Air France flight leaves Papeete at 8:20AM arriving LAX at 7:15PM (same day). You will need to be at the airport by 5:30AM to get through security (long, slow lines) to make the flight.
The cruise line knows all of this and will make sure you are off the ship and on your way in plenty of time.