Deep Neural Networks: Are they able to provide insights for the many-electron problem or DFT?

"However, it is notorious due to the exponential wall"

That is completely true, though there's indeed some methods such as FCIQMC, SHCI, and DMRG that try to mitigate this: How to overcome the exponential wall encountered in full configurational interaction methods?. The cost of FCIQMC still scales exponentially with respect to the number of electrons when all other variables are treated as control variables, while DMRG scales polynomially in the number of electrons but exponentially in something else (called the "bond dimension"). So while there is probably always an exponential wall, the wall is not always the same wall, and it can take a lot longer to hit one wall than the other in many cases and vice versa in other cases.

"for example, see section II (C) of Walter Kohn's Nobel lecture"

There has been some discussion about what Kohn said here: Was Walter Kohn wrong about this? (this is not 100% related to what you're saying, but it's related).

In fact, it is the Kohn-Sham density functional reformulation of quantum mechanics that lays the foundation for the current matter modeling.

That is true for some of the "current matter modeling" going on. There is also some matter modeling such as my entirely ab initio prediction of the carbon atom's ionization energy to within 1 cm$^{-1}$, and everything in here: How accurate are the most accurate calculations?, and everything in here: Are there examples of ab initio predictions on small molecules without the "major approximations"?, and this: High precision helium energy, and quite a lot more in the realm of matter modeling, for which people will run as far away from you if you ever mention DFT.

"A recent study published in nature chemistry claim that the deep-neural-network method numerically solves the electronic Schrödinger equation for molecules with up to 30 electrons with Quantum Monte Carlo methods."

Such studies are interesting, but we did 54 electrons both here and here.

Can deep-neural-network offer similar insights

Probably the number one criticism I've heard from machine learning experts is that while deep neural networks can give extremely impressive results, they usually do not give insight the way a physical theory does. For example, Yuri Boykov, a prominent expert in computer vision, told me that in person last year.

or solutions about the solution of many-electron Schrödinger in the framework of density functional theory (DFT)?

They can surely reproduce the same "solutions" within some margin of error, though this is typically for systems similar to those on which the neural network was originally trained, likely even more so than density functionals working best on systems for which the functionals were optimized, because even on completely different systems, at least the functionals typically have a lot of known physics built into them, see for example this: Mathematical expression of SCAN (Strongly Constrained and Appropriately Normed) constraints in DFT, whereas neural networks are not born knowing anything about physics, chemistry or any theory of matter, though they do learn impressively quickly.

Such as the finding of universal energy functional defined by Kohn-Sham's theorem?

Let's keep our pants on for a little bit longer ?.