It would be highly unlikely to have all 3. There is always one entity which explains most of your symptoms. Psoriatic arthritis can have a rheumatoid arthritis like presentation in 15% of the cases. Ankylosing spondylitis can also be accompanied by arthritis of the small joints. While psoriasis does have great weight when making a diagnosis of psoriatic arthritis it is not an absolutely necessary criterion.
Drugs from the class of anti-TNF inhibitors are FDA approved and work well for all these 3 diseases.