What are possible causes of severe wrist and joint pain?

Several. Trauma (usually obvious to the patient), infection, rheumatoid arthritis (usually more than one joint), gout (usually just one joint), pseudogout, dequervain's tendinitis, and carpal tunnel syndrome are some possibilities. Other conditions generally cause pain that is not severe.
TRAUMA & ARTHRITIS. Wrist pain can be caused by: trauma, such as sprains and fractures, can usually be diagnosed by history, examination and x-ray- although MRI or ct scan may be necessary. Arthritis can be the result of an old injury or an inflammatory condition (rheumatoid, lupus). Some people are born with anatomic variations that predispose them to pain. See a hand surgeon for a comprehensive evaluation.
3 Big Ones. I think you mean pain without any injury? While many things can cause "pain" in the wrist, i see 3 common causes "severe" pain in the wrist (assuming no injury occurred): gout, bad cases of carpal tunnel syndrome, and de quervain's. Pain from other causes may be described as "bad", but these three causes seem to make patients miserable with pain that's "killing" them! see your dr!