The first name of Danny John-Jules's character is Dwayne, which echoes the alter-ego of his long-running Red Dwarf (1988) character Cat, Duane Dibbley.
Some streaming services inexplicably cut the episodes from about 55 minutes, down to the more usual 42-ish minute viewing time, meaning a lot of the viewers have completely missed several hours of the series.
If you have only seen the 42 minute versions, it's worth finding the uncut episodes.
The DI's shack is in fact a temporary structure, rebuilt for every series.
The actual location of the shack is Anse de la Perle beach, just outside Deshaies (the town that plays Honore), and is disassembled at the end of filming, and put into storage for the winter.