Well, I'm not a vet, but I work with the local shelter doing placements and foster to many cats at home (too many! Argh!!!).
Seriously, I've been doing this with my wife for years.
It sounds parasitical to me. Rabies??? I haven't dealt with that really, but I don't think an animal with rabies can live for a year...and mange usually involves a fur/skin issue as well.
10 is old, but not that old. Then again, street cats have shorter life spans, so it is possible. If you haven't taken him to a vet, then I suppose money is an issue.
Try getting whatever over the counter wormers you can get and treat the cat with that. If that doesn't fix it, you'll need a vet to diagnose the parasite.
The coughing up... are you sure it isn't hairballs? Try a hairball remedy fourmula from the store. There's all types, even dry foods that can help. Brushing a cat doesn't help much, although they tend to love it! So do it with a fine comb if you'd like. But cat hair is endless, the specialize in shedding. Ket me know what happens. I haven't been around, but I'll make more of an effort.
I am a vet and this generally means that you cat may have mange. It could however mean that the black cat could have been infected with rabies and you not know it. Alot of times when theses things happen its best o tale it to the vet right away they can prescribe antibiotics to give. Sometimes however old age does have its side affects such as laziness with the litter box and loneliness. But like I said take him to your vet to see what they can do for you and your cat!!!