Only electron in the universe, does it produce magnetic field? Imagine there is only 1 matter in the universe, a free electron. we can't tell if it is moving or not so does it produce magnetic field? I know it has an intrinsic spin but it is just a quantum value so we can safely ignore.
 A: The electromagnetic field of an electron is described by an object called the field strength tensor. So for your lonely electron we would simply calculate its field strength tensor (which would be pretty simple for an isolated point charge).
The electric and magnetic fields are in effect views of the field strength tensor from different perspectives, and to take these views we have to define the reference frame that we are using. For example if we choose the rest frame of the electron then the field strength tensor looks like a purely electrostatic field. If we choose a reference frame in which the electron is moving then the field strength tensor will look like a combination of an electric and a magnetic field.
So the question of whether the electron has a magnetic field is not a meaningful one. It's like asking whether the electron has a velocity. Since velocities have no absolute meaning we can only ask about the electron's velocity relative to some reference frame. Likewise we can only ask about the electron's magnetic field relative to some reference frame. Even in a universe that is completely empty except for the electron there is nothing to stop us specifying reference frames in which the electron can have any velocity we want (less than $c$ of course :-).
A: It is inappropriate to ask "does it produce magnetic field?" since one can always go to electron's frame of reference (which, in absence of an external force, would be an inertial one) and make the magnetic field zero.
A more well-posed question would be: Does it produce electromagnetic field?
It is important to understand here that the mechanism of observation inherently involves an interaction with the system. To observe an electromagnetic (EM) field, for instance, we observe the motion of a test charge interacting with the field.
So if there is no other particle in the universe, there is no interaction, and therefore, to ask if there is something like the EM field is absurd. 
I remember Feynman and Wheeler address a similar issue in their work on the Absorber Theory
