IMO.....
No, you don't have to cut them away. The dead part is now "live rock" so to speak, and if kept healthy, the coral will grow again.
Low water flow, place it on the bottom if you have metal halides. They send out some nasty stinger tentacles at night/feeding time. Do not put other corals right by it. Stinger tentacles are hard to see at night, as they are translucent (thin, but long).
I would guess, like other polyps, they can be fed, but I don't think its necessary. The coral itself will absorb nutrients from the water column, and will catch whatever foods are leftover from your fish feedings.
Best,
Ilham