To make the lime syrup, place the sugar and water in a small saucepan and bring the mixture to a boil, stirring until the sugar has fully dissolved.
Pour in the lime juice and return the syrup to a boil briefly. Transfer it to a bowl and leave it at room temperature for at least 2 hours, or up to 6 hours, so the flavor can develop.
Heat the oven to 350°F. Grease a loaf pan, line it with parchment paper, and set it aside.
In a small bowl, whisk together the flour, baking powder, baking soda, and salt until evenly combined.
In the bowl of a stand mixer, beat the eggs and sugar on medium-high speed for about 3 minutes, until the mixture looks pale and well blended.
Add the oil and continue beating for another 3 minutes to help create a light, well-emulsified batter.
Mix in the coconut flakes, lime zest, coconut flavoring, and lime juice. Add the dry ingredients gradually and stir only until incorporated.
Fold in the shredded zucchini gently with a rubber spatula, making sure it is evenly distributed without overworking the batter.
Transfer the batter to the prepared loaf pan and smooth the top. Bake on the center rack for about 45 minutes, or until a toothpick inserted in the middle comes out clean.
Remove the cake from the oven and use a long skewer to make roughly 15 holes across the surface. Spoon the prepared lime syrup over the warm cake in stages, allowing it to absorb before adding more.
Let the cake cool in the pan for 30 minutes. Carefully remove it and place it upright on a wire rack to cool completely.
For the glaze, whisk together the powdered sugar, coconut cream, lime zest, and lime juice until smooth and free of lumps. Drizzle the glaze over the cooled cake.
Finish with a scattering of shredded coconut, if using, and serve once the glaze has set.