Aiptasia doesnt split or "fork" as you say. They are a pest anemone. When they feel threatened they will expell spores that reproduce. They reproduce extremely fast and will kill coral and fish.
I have used Aiptasia X with good results. I have also used pepermint shrimp with good results. Your tank is too small for aiptasia eating fish such as file fish and copperband butterfly.
There is also an invert called Bergia Nudibranch. They will eat aiptasia and can get into the small areas that fish and shrimp can not get into to eat every last bit of them.
http://berghianudibranch.ca/
http://reeftown.com/
If it is only on the one rock...I would remove it. Let it soak in a kalkwasser/fresh water mix for 24 hours, then fresh water rinse it and bake it till it is comepletely dry. Let it soak in ro/di water for a week or so, rinse it real well and then put it back in the tank. Keep a good eye on it because they can still come back.
I believe there is a local breader, but not sure of who it is at the moment.
The Green stuff is definately a macro algae. Cant tell what type it is form the picture. Is it a smooth stem or is it feathery? I looks like it might be evasive. You may want to remove it too.