Can overall joint pain, "double jointedness", and TMJ pain/clicking be from some sort of arthritis or other joint disorder?

Yes, need exam. Arthritis, injuries and other diseases that effect joint can also occur in the jaw joint. The lax ligaments that cause double jointedness can occur in the jaw, especially in women. You should see a dentist who is well trained in treating TMJ disorders for a complete evaluation and treatment. Joint dysfunction is progressive so treatment should start at your age, not wait until severe pain.
Double jointedness. Many people have genetic disposition to be loose-ligamented....Or laxity of ligament. That is being more stretched than most. The pain and clicking comes from the disc of the TMJ being loosely connected due to the stretched ligements and therefore moving around...Often too far and getting into an abnormal position. The click is the return to normal when opening and falls "off" the condyle closing.
Hypermobility assoc. Hypermobility may be associated with a number of causes, and can itself cause joint pain. Rheumatologic evaluation indicated.
Possibly. Systemic Joint laxity (hypermobile joint syndrome) is associated with TMD. Have your TMJ managed by a TMJ specialist.