Should I discard initial MCMC steps before calculating R-hat and effective sample size when starting far from equilibrium?

Question

I'm running a Monte Carlo simulation of a molecule, where the initial configuration is far from the equilibrium state. I run multiple MCMC chains in parallel, and after a certain number of steps, the ensemble average of the energy approaches the equilibrium energy.

To assess the convergence of my chains, I intend to calculate the R-hat statistic and the effective sample size (ESS). However, I understand that the early steps of the chains may not reflect the equilibrium distribution due to the initial configuration being far from equilibrium.

Here are my specific questions:

1. Should I discard the initial MCMC steps (burn-in period) before calculating R-hat and ESS, given that the initial configuration is far from equilibrium?
2. If so, how many additional steps (after discarding the burn-in) should I use to reliably calculate R-hat and ESS to judge convergence?
3. I am using the criterion that the ensemble average of the energy has reached a stable value to determine when the burn-in period has ended. Is this an appropriate criterion for determining the end of the burn-in period, or should I consider other metrics?

Any guidance or best practices would be appreciated!