Number of electrons in an orbital How do we know number of electrons per element since electrons do not have shape and volume? Isn't an electron just quantized fluctuating probability wavefunction? Is there an experimental study supporting the idea that electrons are the moving particles in orbitals?
 A: When we solve the hydrogen atom Hamiltonian, we get quantised energy states that are allowed for an electron. These states correspond to the wavefunction of the electron and are called orbitals. And since these orbitals are stationary states, the number of electrons in the ground state is constant. So when you say:

Isn't an electron just quantized fluctuating probability wavefunction? Is there an experimental study supporting the idea that electrons are the moving particles in orbitals?

The quantised fluctuating wavefunction itself is what we call orbital. However it is to be noted that they are exact only for hydrogen atom. This is because an orbital inherently doesn’t involve electron_electron interaction. Spectral lines provide indirect information about the presence of orbitals. But there have been direct evidence of quantised states by the means of orbital tomography, however the validity of orbitals has been discussed here. 

A: 1) "How do we know number of electrons per element since electrons do not have shape and volume?"
Answer: At the very beginning of the 20th century there were experiments in which the scattering cross section of elements were measured, in Classical Electromagnetic theory we can calculate the cross section of the one single electron .  And in these experiments it was obtained that the actual cross section of some element is just an integer number multiplied by one single electron cross section. So it is a reasonable idea to interpret this integer number as the number of electrons in this particular atom.
2)"Isn't an electron just quantized fluctuating probability wavefunction? "
Actually the wave function is not the electron itself, it is a very handy tool to calculate the probability to find an electron at some particular point
3)"Is there an experimental study supporting the idea that electrons are the moving particles in orbitals?"
The only experimental basis was a Rutherford experiment, in which in occurs that electrons are somewhere around the nuclei, from the classical point of view, the only avalible pattern of motion is orbital motion. Speaking about the experiment,  actually you can measure that the orbital momentum of the Hydrogen's electron at the ground state is zero, from the classical point of view it means that there is no orbits, actually. So, we can say that the idea of orbitals does't work....but just because the classical mechanics itself can't be applied to atoms.
