What disease could cause the following symptoms: bleeding gums, loose teeth, painful and swollen joints?

Vitamin C deficiency: The symptoms you described are classic symptoms of vitamin c deficiency(scurvy). The symptoms in part or in whole could be secondary to a number of medical problems. Without more clinical information a more specific response is not possible.

Scurvy: It is not likely in this day and age, but your symptoms are ones sailors on long voyages used to suffer from. It is a lack of vitamin c. An aside, british sailors (and now all uk citizens) are called limey's since they warded of scurvy by eating limes on long voyages. Probably unhelpful, but historically enlightening.