Can hypermobility cause pain in the calf muscles?

i'm not exactly sure what you're referring to when you say hypermobility? But often it's used to indicate a joint can be extended more than average. overuse of the calf muscles can cause pain, for example going for a walk or run longer than usual. Hypermoblity around knee joint can also cause some pain in calf (as well as thigh) muscles. So in either case, yes you can get calf pain.

Strictly speaking, no. Hypermobile joints are present when ligaments are naturally more lax in some people which may relate to variations in their collagen genes that allow them to stretch more than those in other people without joint hypermobility. Calf pain can occur for a variety of reasons including tight calf muscles that need to be gently stretched and relative overuse (sore muscles).