Almost any tissue or organ can be "harvested" and transplanted. Common tissue and organ transplants are the cornea of the eye, skin, bone, muscle tissue, lungs, heart, liver, and kidneys.
When you ask about the forearm or leg, even though this is technically possible, it is extremely rare to take a whole leg for example and transplant that leg onto a recipient. But, what is more common, is to take a bone out of a donor, say the femur, and transplant that bone into a recipient.
Organ and tissue transplantation in general is fraught with moral, religious, and ethical dilemmas. Donation is a highly individualized matter that should be discussed by the patient, his family, and the patients physician. Be advised that if you wish to donate your organs after death your surviving next of kin must consent to this after you die. If they do not, the donation will not take place regardless of your wishes.
Yes, limb transplants do take place, and though they are not terribly uncommon, the ethical ramifications are still being debated.
Also, like with all transplants, there is the risk of rejection so patients who are to undergo a transplant are given anti-rejection medication and immunosuppresive therapy to decease the risk of rejection. With immunosuppresive therapy transplant patients are at an increased risk of cancer, infections, and other disorders because as the therapy name says, the immune system is somewhat suppressed. Rejection during limb translation is unfortunately a reality for a quite a few patients.
To lessen the risk physical therapy is very important after transplantation.
Furthermore, there is even a psychological risk with transplantation (this can sometimes be seen with heart translates as well) where a person begins to associate with the deceased person. For this reasons psychological therapy after transplantation is very important, not only to avoid dissociation between a person and the person's new limb, but also to counsel in the chances of the limb being rejected.
Yes, this is possible. It would be very rare to find a whole limb that would "fit" the patient. Both internally and externally.