equivalent way(s) of expressing P=?NP problem in linear programming?
This answer is for your "somewhat related question":

In the following recent paper, Klee-Minty cubes were used explicitly to show that  there exists a pivoting rule for the simplex method (not one of the standard ones, like Dantzig's pivoting rule analysed by Disser and Skutella) for which it is PSPACE-complete to decide whether a variable enters the basis for the simplex method with this pivoting rule. It was left open (Conjecture 3 in the paper) whether the same result holds for Dantzig's pivot rule.  

I. Adler, C. Papadimitriou, and A. Rubinstein (2014).
On Simplex Pivoting Rules and Complexity Theory.
In Proc. of IPCO, pp. 13-24.

In the following paper we prove that conjecture and strengthen Disser and Skutella's result to show two PSPACE-completeness results for the simplex method with Dantzig's pivoting rule (one for whether a variable enters the basis "along the way" and another for whether a variable is in the final solution found by Dantzig's pivot rule).

J. Fearnley and R. Savani (2015).
The Complexity of the Simplex Method. In Proc. of STOC, pp. 201-208. 
http://arxiv.org/abs/1404.0605

We use the known connection between linear programs and Markov decision processes (MDPs). A central part of our construction are the exponential-time examples due to Condon and Melekopoglou for single-switch policy iteration for MDPs. Those examples use a reflected binary Gray code, as used for Klee-Minty cubes. We use a modification of those examples as a clock to drive iterated circuit evaluation in our construction.