As far as I can tell, this paper has nothing to do the generalized solution to the Navier-Stokes equations. It deals with flows near surfaces (boundary layers) and not flows in general. After skimming the paper I could only find one mention of the Navier-Stokes equations, and that was just for a channel flow identity derived from the Navier-Stokes equations under very particular boundary conditions (so not general at all).

This paper provides an interesting connection between the spectral properties of turbulence and the second-order moments of velocity in boundary layers (the "spectral link" as the title says). That is interesting to experts but it is a bit too abstract for the general public to understand easily, unfortunately.

No. This is a better approximation for boundary layer turbulence, it's not a golden equation. Better approximations for annoying problems in fluid dynamics are published quite often. I think has plenty of practical use though, I'll probably be trying to integrate it into some cfd work to see how it behaves compared to our current approach at work. Disclaimer that I am not a physics phd, but I do work with a hydraulic research group so I've picked up bits and pieces.

If you want to see a fun example of fluid dynamics approximations, look at the Wikipedia pages for the Darcy friction factor and Colebrook white equation.

I would also like to know that.