Yes doping plays a significant role in electron/hole mobility. In terms of temperature, mobility is affected by scattering. There are 2 types of scattering that affect speed: impurity scattering and lattice (phonon) scattering. Impurity scattering is due to just that, doping of the semiconductor. In lattice scattering, an electron is scattered within the material due to vibration of the lattice (which increases with temp).
A complete graph of mobility vs. temperature would look like a carrot top ^ ..... With impurity scattering dominating at lower temps, mobility increases with increased temperature. At a certain point, lattice scattering dominates at much higher temps. Then it's the opposite, mobility decreses with increasing temperature. The Einstein relation I mentioned previously was assumed valid when lattice scattering is dominant... . the usual area of operation.