If I recall correctly, the original MM idea was to determine if the motion of the earth, relative to the presumably stationary aether, made any difference in the transit time of the light beams. The motion of the entire solar system, and the galaxy, etc., would be common features to any such measurement, and could thus be ignored.
Yes.
Is the electric field of this moving charge (that is, current) in the direction of the current density vector?
No. In fact, Lorentz contraction causes the electric field from a moving charge to become weaker along the axis of motion and stronger in the transverse direction. The field becomes compressed along the axis of motion.
If you think carefully--very carefully--about this, it might seem that this gives rise to a situation that is not Lorentz invariant: as the charge passes by a "stationary" test charge, a stationary observer should expect the test charge to be subjected to a greater electostatic force than would be measured by an observer that is comoving with the moving charge. What's a test charge to do?
As it turns out, this difference in force is exactly compensated by the magnetic field that is observed by the stationary observer, created by the motion of the moving charge. This field is not observed by the comoving observer. They agree on the net motion of the test charge.
In other words, the magnetic field is the manifestation of effect of Lorentz contraction and time dilation upon an electric field. It's relativity you can play with at home.
Or is there no electric field of the charge in motion because it is mechanically forced to move?
The total electric field around the charge is invariant. Gauss's Law, once again. There is no additional electric field that is created by the movement of the charge. Battery, black hole, or baseball bat, it doesn't matter what causes the charge to move. The divergence of the field equals the charge density, end of story.
I'm confused because there is no relative motion between the observer and apparatus.
Then that's the frame you use for that observer. The issue is that a moving observer will also have to observe a null result for that apparatus. It has to work in all possible reference frames. The different observers aren't entitled to disagree about the outcome of the experiment; either the interference fringes shift or they don't. They will, however, disagree about the size of the interferometer.