Mobility depends more on the effective mass of carriers in the semiconductors, various scattering phenomena associated with the crystal. For intrinsic semiconductors mobility does depend on the carrier concentarion which in turn depends on band gap and temperature.
Not really. Diamond (Eg ~ 5.5 eV) has awesome mobility (~1450 cm2/Vs).
However, wide bandgap materials show less leakage (in the off-state), as the thermal carrier generation is slower (because of the large bandgap), which reduces loss.
The effective mass of electron has a direct relation with Energy band structure, which is m*=1/(1/h2)(d2E/dK2), where h is the simplified h factor, d2E/dK2 is the second order differential of the E with K, and this should come from the band structure which you would like to know.