Electrons, magnetism and electromagnetic waves I know that moving charge (such as electrons moving around nuclei) produces a magnetic field. I also know that moving charge (again, such as electrons in atoms) can produce electromagnetic waves, which are just disturbances in magnetic and electric fields. So, what is the microscopic explanation of the electrons' motion that allows them to produce both variations of the magnetic field (one with disturbances/ripples and one without) independently and simultaneously?
 A: I know that moving charge ... produces a magnetic field. It produces electric and magnetic field.
"What is the microscopic explanation....". I am afraid it all boils down to Maxwell's equations. One can of course try to come up with analogies, and wave hands, but I have seen enough exceptional cases to know that analogies will always be subpar.
Following things may help:


*Avoid talking about electrons moving around nuclei. It is best to sort out the classical electrodynamics first before going to what happens in quantum mechanical setting.


*Electric and magnetic fields should not be viewed as separate. There is single electromagnetic field. How much of it is electric and how much is magnetic is a matter of perspective. It is not completely arbitrary, but it is nevertheless true that if I am stationary and you are moving at constant velocity relative to me, then the field that I perceive as fully electric you will see as mixed electric and magnetic.


*The main difference in motion of charges, for the purposes of producing electromagnetic field, is whether the charge is accelerating. Acceleration means that the velocity, a vector, that characterizes the motion of the charge is changing in time. Note that moving in a circle is a special kind of acceleration where magnitude of velocity (speed) stays fixed, whilst it's direction changes. Accelerating charges can produce electromagnetic waves (see Liénard–Wiechert potential).
Perhaps the following will help.
Finding the field produced by non-accelerating point charge


*Find an inertial reference frame in which this charge is at rest.


*Solve electrostatic Maxwell's equations, or simply sub-in the Coulomb's field


*Boost the field into the inertial frame that you care about
Finding field produced by accelerating point-charge
Use some version of Liénard–Wiechert potential. This can get quite complicated depending on the trajectory of motion etc.
General
Define the motion of your charges and solve Maxwell Equations
A: The Jefimenko equations are great for this; thy show:
$$ \mathbf{B}(\mathbf{r}, t) = -\frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^3} \times \mathbf{J}(\mathbf{r}', t_r) + \frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^2} \times \frac{1}{c}\frac{\partial \mathbf{J}(\mathbf{r}', t_r)}{\partial t} \right] \mathrm{d}^3 \mathbf{r}' $$
The first term gives you the non-radiative magnetic field that goes to 0 quickly as you move away from the source (like a normal current or moving charge).
The second term gives you the radiation field, i.e. "waves/disturbances".
To examine how this works with a point charge, the current $\mathbf{J}$ for a point charge is basically just $q\mathbf{v}$, and thus $\frac{\partial \mathbf{J}}{\partial t}$ is just $q\mathbf{a}$. That fits with the general rule of thumb - radiation is caused by the acceleration of charges while magnetic fields in general are caused by moving charges (accelerating charges are also moving, so as you'd imagine they create both)
A: Imagine you are standing in front of an orbiting electron looking at it. The plane of its rotation is horizontal, and when it passes closest to you, it goes from your left to your right.
Electric field
As it gets closer to you, the electric field increases, because electric field made by a charge is proportional to $\frac{1}{d^2}$ where $d$ is distance. It’s distance from you is changing. From max field to next max field takes $t$ - the time to go around once. When it is far away, the positive nucleus is closest to you, so the field is the opposite sign.
The field direction is at you or away from you.
Magnetic field
A moving charge makes a magnetic field. With your thumb right along the vector of motion, your fingers show direction of field which curls completely around the path of the charge. When the charge passes closest to you (left to right), this field points down. When it passes furthest from you (right to left), it points up. When going away from or toward you, it is zero. Max field to max field is $t$.
The field direction is up or down.
A: 
I know that moving charge (such as electrons moving around nuclei) produces a magnetic field.

An electron is not only a charge, but also a magnetic dipole. The electron has its intrinsic magnetic field independent of a movement around the atomic nucleus.
The formation of a macroscopic magnetic field has to do with the alignment of a large number of magnetic dipoles of electrons. Since these are chaotically distributed in some materials, the magnetic dipoles of the electrons are not noticeable in this situation.
However, due to a more or less strong external magnetic field, the magnetic dipoles of the electrons are aligned and can then be measured as a macroscopic field.

I also know that moving charge (again, such as electrons in atoms) can produce electromagnetic waves, which are just disturbances in magnetic and electric fields.

Electromagnetic emission of electrons occurs in two cases. First, excited electrons in atoms that fall back to their unexcited level emit photons.
Secondly, moving electrons are deflected sideways in an external magnetic field, and during this deflection the kinetic energy of the electrons is converted step by step into emitted photons.

So, what is the microscopic explanation of the electrons' motion that allows them to produce both variations of the magnetic field (one with disturbances/ripples and one without) independently and simultaneously?

What you describe as ripples are EM waves produced by the synchronous acceleration of electrons in a radio antenna. You have to distinguish between radio waves as a special case of polarised emission of photons from synchronously accelerated electrons and chaotic emission from excited photons. In the latter case, neither a magnetic nor an electric field can be detected in the electromagnetic radiation.
Only in the case of synchronous acceleration of electrons is the EM radiation polarised and a common electric and magnetic field generated. This field can be detected with a radio receiver.
