I have lived here since 2005, not really sure about growing the bananas they just grow out in the back and I go cut a bunch as I find them ripened.These are not full sized bananas, they are about half the size I was used to getting back on the mainland.However the are very yummy!
The papayas grow very fast, but again I havent tried to grow one so I am not really sure

One of the local things though involving the papaya is to cut it while green, thinly slice it and pickle it in vinegar and hot peppers.It is actually very good. It is also used as a local remedy to cure all kinds of skin problems including ringworm.
Some of the local dishes that are popular :red rice, chicken keleguen,anything with spam, theres a ton just using coconut and I would be happy to send you any recipes I come across.
I had a talk with my neighbor who has lived here her whole life and she actually took me for a walk and showed me other stuff, sugar cane, taipioca,and about 6 other fruits that I cant for the life of me remember the name of atm.