If I am hypermobile, or flexible, does that mean I have Ehlers Danlos Syndrome?

Not necessarily. Hypermobility is one of many associated characteristics of Ehlers Danlos Syndrome. Simply being hypermobile does not mean that you have Ehlers Danlos Syndrome, though hypermobility is a necessary component of the physical examination used to establish a diagnosis of the most common form of Ehlers Danlos Syndrome, the hypermobile subtype (hEDS).