its a skeleton of a brain coral, it will not harm your water if its just a skeleton. But if it still has dead tissue on it then yes, this will harm your water. I had a brain coral die on me, and i placed it in my frag tank, its now fully grown back to its original size, and has a completely different color. I got lucky but it was achieved by giving it more light before all the tissue was completely gone off the skeleton. Your's looks like theres nothing left but you might still have some luck.
the rock is Base Rock I'm guessing. It will take time but it will soon get life on it such as coraline algae and other encrusted critters.
as for the filter, you can use the same black sponge but it needs to be rinsed free of debris monthly. I use a small amount of mixed water to keep as much bacteria as possible. Carbon and any other chemical media should be replaced at this time as well. Water changes once weekly or every other week. What ever you prefer, just make sure your water levels stay consistent.