It's the convective layer that is differentially rotating, everything outwards from ~0.7 Rs. The part of the sun that appears to be "rigidly" rotating is the radiative zone, the part that lies between ~0.2 and ~0.7 Rs. The reason for this is not known indetail, but I suspect the answer is rotational shear working on the convective zone. Since the rotation is axial, but the convection is radial, you can readily see that the convective zone can shear into horizontal motion that depends on the latitude. See, for instance, Rotation of the solar interior, a chapter from the book Dynamic Sun, or perhaps Helioseismology, which appeared in Reviews of Modern Physics. Or look up The Internal Rotation of the Sun, which appeared in the 2003 issue of Annual Reviews of Astronomy and Astrophysics.

I think, as do most, that the solar magnetic field is driven by motions in the outer, convective zone of the sun, especially near the tachocline, which is the boundary between the lower radiative zone, and the upper convective zone. Large-Scale Dynamics of the Convection Zone and Tachocline is in Living Reviews of Solar Physics, and should be visible without a subscription. Or see the Solar Dynamo page from Marshall Space Flight Center.

I doubt it. The core/mantle boundary in Earth is a sharp phase transition, where the outer core is liquid, and the mantle is "solid" (it's actually more "plastic", you might think of it as an extremely viscous liquid). There shoul not be a phases transition like that, so deep inside the sun.