Couple clarifications - caulerpa is a type of macroalgae, what you have is not that. You do have a kind of macroalgae there, technically its not a plant though a lot of saltwater macroalgae look like them. Not sure on exact ID for your macro at this time. Can you post a close up pic of one of the branches?
If you look at the macro, are there a lot of pods on it? Some larger amphipods can actually eat macros like that to the point of doing serious damage.
Turning green is not a good sign. Do you know what your phosphate levels are? Have there been any temperature or alkalinity swings recently in your tank?
On more stiff macroalgae like that, the white dots are most likely spirorbid worms or similar. They won't cause problems directly, just annoying and don't look the best.