Welcome! Wow, there's a lot of questions there!!

First, I have no idea what Anse Heureuse is, but I assume someone else here does know.
As far as what to leave in the car--NOTHING!!! Not at any time, at ANY place, for ANY length of time. Period. EVER. Got that??
Regarding where to park, if you're talking about at the beaches, most of the beaches have really easy parking. As far as taking local busses, they are generally pretty unreliable and if you're talking about at night, forget about it!! As far as grabbing a cab, probably your best bet would be to hook up with a cab driver early on and use him or her as your go-to person, as if you're looking primarily to catch cabs at night, they are often few and far between. You can usually get a restaurant to call you a cab, but better if you hook up with someone ahead of time, like TTOL sponsor Wilfred Brooks. Click on the TTOL sponsors tab and scroll down for his contact information.
As far as bicycling, I would say NO. Drive a car or take a cab, as the roads are often not in great shape and you don't know the roads either. Put a couple thousand pounds of metal between yourself and harms way.
Other people will chime in on some of the other questions.
Enjoy your trip!