Does the electromagnetic field "spin"? Due to electron "spin", a small magnetic field is produced.  Maxwell's equations imply that magnetic fields are due to changes in electric fields.  Is the magnetic field produced then because the electric field is "spinning" with the "spinning" electron, in the quantum sense of "spinning" and this change in electric field is generating the magnetic field?
Can one generalize to say that the magnetic field thus would spin when a magnet is spun?
 A: It is certainly possible for an electromagnetic field to spin and this can be demonstrated by producing such a field in a cavity resonator. The electromagnetic field in a resonant cavity is normally stationary but varying in amplitude so that the changing magnitude of the electric field produces a varying amplitude magnetic field which in turn produces the varying electric field. When a spinning mode is produced the amplitude of the electric and magnetic fields are constant but it is their rotation about the axis of the cavity which produces the time variation required to sustain the fields.
The field equations for the spinning electromagnetic field can be derived from those of the conventional stationary field. They satisfy Maxwell's Equations and the Poynting vector can be shown to point in the direction of field spin. Computer modelling of the propagation of the fields using the FDTD (Finite Difference Time Domain) technique clearly show them to spinning. Also a practical experiment has been done to confirm that measurements of the spinning electromagnetic fields are as predicted.
Further details are available at http://mike2017.000webhostapp.com/
 which includes field plots of the spinning fields.
A: If by "spin", you mean rotate around its axis, like the earth does every 24h, then it is incorrect that the electric field of a point particle spins. A point particle doesn't have dimensions so it doesn't have an axis to rotate around and thus no magnetic field is produced.  
The property of "spin" of elementary particles is not caused because of their rotation. 
Now, the magnetic fields are different because there are no magnetic monopoles, so a magnetic field does rotate when the magnet is spun. 
A: Spin corresponds to quantized angular momentum. However a substantial fraction of the spin angular momentum of an electron is included in its surrounding electromagnetic field where a nonzero Poynting vector does exist everywhere outside of its spin axis. This electron-bound Poynting vector  corresponds to electromagnetic energy-momentum density circulating around the electron.  The local magnetic field at a given point is given by the electron’s dipole field while the electrostatic field results from the Coulomb-field of a point-like charge [1].
Please also note that neither an electrostatic field nor a magnetostatic field can rotate like a rigid body. This misconception would contradict Maxwell's and relativistic electrodynamics. 
See Spinning magnets and Jehle’s model of the electron.
A: Your statement "Maxwell's equations imply that magnetic fields are due to changes in electric fields." is not complete.
A corrected statement is that Maxwell's equations imply that magnetic fields are due to changes in electric fields AND due to currents (which can be stationary):
$$
\nabla\times\mathbf{H} = \mathbf{J}+\partial\mathbf{D}/\partial t
$$
As you can see, the magnetic field $\mathbf{H}$ has two "sources": the $\partial\mathbf{D}/\partial t$ part is due to varying electric fields as you said (where $\mathbf{D}$ is the electric displacement), but the $\mathbf{J}$ is the part due to the free currents. This is why a coiled wire with a constant current running through it creates a magnetic field (without need of changing electric fields).
In the case of the electron spin, this goes beyond my areas of knowledge, but according to my limited understanding of quantum mechanics, the magnetic field comes from the stationary particle current associated with the wave-function of the electron. So it is similar to the magnetic field arising from a coil of wire with a current.
As a further related note: interestingly, Maxwell's equations apply to any inertial frame, so you could argue that an observer moving with respect to the electron will see a changing electric field (because the electron is moving), and this will create a magnetic field which apparently does not exist for a stationary observer. This is because different observers will not agree on the electric and magnetic fields separately, however they will agree on the existence of an electromagnetic tensor (which includes the electric and magnetic fields as its "parts"), and they will agree on the physical effects produced by it.
