Can having dysthymia qualify you for disability?

I doubt that. The diagnosis of dysthymia would qualify you to receive disability. The reason is that while dysthymia can really impact your quality of life - it should not prevent you from basic functioning. Disability is based on impairment not degree of sadness or dissatisfaction with oneself or their life.
Depressed. No, probably most of us suffer from dysthymia, if it gets into severe depression untreatable with available meds then that is different but I think no matter how depressed you are disability is not that easy to get.