You can leave the skeleton. It's up to you. Live rock from the sea is often just dead coral skeletons. Those tubes are part of the sponge. Nothing to worry about.
Coraline algae comes in many colors. Purple is the most common.
As for bleaching. That can be caused by a number of factors but usually too much light or too much light too quickly. The rock they are on is probably higher up than it used to be. I would move it down.