Electron configurations beyond hydrogen In 1990, the Journal of Chemical Education (American Chemical Society) published a little bit controversial article titled "The nature of the chemical bond—1990: There are no such things as orbitals! J. Chem. Educ. 1990, 67, 4, 280" Article link. The author has mainly objected that orbitals are not physical objects, but mathematical constructs, and perhaps chemists take them too literally.
I have also wondered that Schrodinger's equation is solvable for 1-electron atoms, yet chemists have extended the electron configurations to large and heavy elements like lead or mercury. So we have the Aufbau rule and Madelung's rule, and so on to estimate the filling of orbitals. Mullikan had coined the term orbitals as a short form of one-electron wavefunctions, but obviously, all the periodic table elements are not one electron atom after hydrogen. Nevertheless, the orbital concept has been extended to larger atoms even with hundreds of electrons.
The question from the point of view of physics is if you happen to see the electron configuration of Pb as [Xe]4f$^{14}$5d$^{10}$6s$^{2}$6p$^{2}$, what is the experimental evidence that lead's electron configuration is like that and this is the correct orbital order. Note that Bohr came up with electron configurations based on the chemical properties of elements.
I have never seen any chemistry or physical text that shows or discusses experimental evidence of electron configurations. The shell structure of atoms is indeed mentioned via X-ray studies. What is the experimental evidence of orbitals and orbital filling orders in heavier elements beyond hydrogen?
 A: 
I have also wondered that Schrodinger's equation is solvable for 1-electron atoms, yet chemists have extended the electron configurations to large and heavy elements

I absolutely agree with you. Schrödinger's equation can only describe the excited states in the hydrogen atom and in hydrogen-like ions. Using this equation to describe the electron arrangement of other elements does not reflect the reality of chemical compounds.
Bound electrons occupy a certain volume in the atom and can be considered immobile. During bonding with another atom, the shape of the volumes changes and the electrons can shift their position a little. This results from empiricism in chemistry. (See this answer.)

So we have the Aufbau rule and Madelung's rule, and so on to estimate the filling of orbitals.

Absolutely fundamental is the Pauli exclusion principle. It states that electrons in an atom differ in at least one quantum number.
Much less evidence-based are the shell regions, which are defined by the use of special Spherical harmonic. The development of these harmonics was as follows.

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*Periodicity of hydrogen spectral lines.

It is a fiddle to get a formula for the periodicity of the emission lines of hydrogen. Balmer was the first to succeed in 1885 and Rydberg made an extended formula out of it. All for hydrogen, mind you.


*Bohrs shell model of atoms

An important piece of the puzzle for Bohr's atomic model was the application of the Rydberg formula. In reality, however, only hydrogen and hydrogen-like elements can be calculated with the Rydberg formula. (See the answer to the Why 2nd Shell can have 8 electrons?.)
For a good overview see the answer to the question In chemical compounds, where does the “magic” come from in atomic “magic numbers”?. It starts with


It was and is until today a empirical fact, that elements with 2, 8 and again 8 electrons fill shells in a way, that these shells are particularly stable chemically.


A: In the experiment here the hydrogen orbitals are explored .

The atomic structures have been modeled with the various approximations of the Schrodinger equation, because it very successfully reproduced the spectra wherever it was checked. This means that the mathematical model was validated. The replacing the term "orbit" of the semiclassical Bohr model with "orbitals" came because the mathematics can only predict probable location of the electron, and the orbitals started to be used to communicate the probabilistic nature of predictions.
You ask:

. What is the experimental evidence of orbitals and orbital filling orders in heavier elements beyond hydrogen?

The fact that orbitals have been seen experimentally for hydrogen, and the fact of the success of the theory to approximate the spectra of multielectron atoms, if we really accept that mathematics is rigorous, validates the orbital language for multielectron atoms. No need for extremely complicated experiments as the one above. That is what mathematics is used for. Do we have to continually prove for each satellite the gravitational predictions? We use them because we trust that the  mathematics works.
