After a fin is damaged or torn, the dead tissues usually turns black. That can be a good sign meaning that the fins are healing. I suggest using aquarium salt (a tablespoon per 5 gallons) and MelaFix by Aquarium Pharmaceuticals to aid with healing. If the fins seem to become infected, then antibiotics may be needed.
Torn fins can result from physical damage (other fish, not in your case; sucked into filter intake; spawning, not in your case) or bacterial problems (fin and tail rot). With the fin and tail rot, the fins will look kind of white, a little fuzzy, and raggety. Physical damage doesn't look infected, at least not initially but bacteria or fungus can take advantage of the damage.
You should try to get your biowheel moving again. Sometimes it won't turn if it's hitting against the lid, algae, snail shells, pleats that have come off the spokes, etc.
You can post a photo here that is already on a web site somewhere. There are many sites that let you put up photos. Once the photo is on a web site, while typing a reply, click on "IMG" and enter the URL (web site address) of the photo. Then, once you "Add Reply," the photo will appear in your post.
As for the algae, my page on aquarium algae is at
http://www.fishpondinfo.com/algae.htmFor a lot of algae to grow, they need nutrients (such as fish waste) and light. Does natural light hit the tank at all? How often do you clean the tank? I suggest weekly 30-50% water changes with gravel vacuumings. That will help if you clean less often.
Good luck!