Is the electron in an atomic orbital moving or not? I have debated quite a lot with friends on this question and the answer is still unclear. 
On hand the electron wave fonction is delocalized on the orbital (let's say the one in the H atom e.g.), so the electron is everywhere on this orbital and is not moving. Related answers have already been provided here and here. For sure, it is clear that electron cannot revolve like a ball around the nuclei as it would lost energy and quickly fall on it, so classic picture cannot be right.
On the other hand, the electron had a momentum $p$, and Heisenberg uncertainty tells us that its position is never well defined. If position is not defined, then can we speak of a notion of displacement, and therefore of velocity, at least for the wave function?
 A: What is an electron?
It is an elementary particle in the standard model of particle physics, which is a very well validated model.
An electron carries the attribute of a "particle" because when not in a bound state it looks like a  classical particle and it moves like a classical particle:


The curly line was produced by an electron that was struck by one of twelve passing beam K- particles in a liquid hydrogen bubble chamber. It curves in an applied magnetic field and loses energy rapidly, spiralling inwards.

When freed from the hydrogen atom it has a measurable momentum and all the other attributes in the standard model table.
Atomic physics also has a very successful mathematical model, that for the hydrogen atom describes it as a complex bound state of an electron and a proton in their common electrostatic potential induced by the charges they are bearing. The model is very successful because it reproduces the Balmer and Lyman series for transitions of the atom out of and into different energy states by the emission or absorption of a photon. This is what can be measured by the atomic system, and note it is the whole atom that is described by the solutions of the quantum mechanical equation. The model does not offer a handle of measuring the electron or the proton individually. It allows to visualize the proton at rest and the probability of finding the electron at an (x,y,z) around the proton as a point in an orbital, and the probability distribution has a shape in space, but in reality one is working with the whole atom, and how the charges of the two participating particles, the electron and the proton, respond to interactions from outside the system: that is what the "measurable" hydrogen orbitals are:
 
This is the probability distribution for the hydrogen atom  wavefunction squared, measured with an ingenious method. 
So it is not possible the measure a velocity for particles in a bound state with each other. One can assume that the energy of the energy level is the energy of the particle and calculate a momentum value, but not a momentum vector.
At the quantum mechanical level it is interactions that can be measured and fitted with models. Once a particle leaves a macroscopic footprint it is in the classical regime.
A: You are applying classical concepts to the quantum world, it does not work like that. As noted above, a definition of  "moving" implies some trajectory or path  is involved. There is no such thing for a quantum particle. 
Keep extending the application of probability to all areas, not just say, the superposition principle, but also the different paths along a particle can propagate,  and I am wary of  even using the word propagate, as it implies a definite track, which is misleading.
You could have a look at this : Path Integral Formalism and this: 
The good news is that the particles that make up a football when you kick it will follow the path you intuitively guess they will, they must do, otherwise classical logic breaks down, the bad news is that to follow how an individual particle gets from your foot to the goalpost is far more subtle. This is one of the few cases where it is easier to deal with large numbers of particles, rather than smaller ones.
Also, electrons are indistinguishable from each other, so if the electron field produces another one, you can't track them. Learning quantum mechanics is a stepping stone to a more accurate representation of the quantum world, and questions such as yours arise because it is difficult not to see micro "things" as scaled down versions of the macro world, they are not.
My point is that our best description of the "real" quantum word (to date) is Quantum Field Theory, which does away completely with any direct connection between the classical world and the quantum world, such as the scenario in your post. 
