Re: Rip out the carpet- go for it.
Right. When you rip out the carpet (at least in your room) you might as well take care of as many things at once as possible. Depending on what sort of flooring you find underneath, you may want to disinfect it somehow (white vinegar kills mold I believe) and then put down some sort of hard flooring, be it hardwood, linoleum, etc. that you can clean easily (depending on what you can afford). Then you should see about the walls and the ceiling, cleaning them if you can. I don't know about wallpaper that may be there, or paint, or painting over (the fumes can be bad) but if you have some toxic wall surface now or mold growing then it may be worth considering. I'll bet there are websites out there that address these things in detail.
Also any air ducts that come into your room, who knows what may be in the air ducts, perhaps you can get them cleaned, put a filter on them, etc. That way when you move things back into the room (cleaning the furniture might be advisable, too) you can have a true clean start. Then things will be cleanable in the future.
When I was 13 we moved to a new house (new to us - 80 year old house) which had no carpet and no curtains and that has worked out really well for us. It's easy to take care of, and you can really get RID of dust and dirt.
On the other hand, my friend is essentially allergic to his house. He lives with his parents. They have pets, one cat in particular likes to sneak into his room, as well as carpet. He is allergic to animals and dust, so he actually feels BETTER when he is out of the house.
Even though the whole mold/carpet thing seems most likely, your symptoms sound pretty serious. If you continue coughing blood or passing blood in your stool you really should see a doctor, at least for his diagnosis even if you choose to use a different form of treatment. It may be something else, or a combination of several things.