The cat cruise leaves around 3pm and gets back around 6pm. It's usually offered on Mondays, Wednesdays and Fridays.
The Beach Party is on Monday nights. There is usually a cocktail reception at 6:30. The buffet dinner on the beach starts at 7:00pm. Crab racing usually starts at 8:15 and the beach party entertainment at 9:15. Then that night they also have the Margaritaville shuttle that leaves around 10:30pm.
Dunn's River is offered every day except Friday. It departs the lobby at 9am and you're back around noon.
No, there are not parties on the Island at all. It closes at 5pm. However, they are now offering private wedding ceremonies on the island after hours.