This indicates very high posterior correlation between some parameters. Have you looked at the pairs plots? How big are ESSs for the quantities of interest?
They don’t look that bad. Running just once is not enough to say those estimates would be bad (based on what you show).
No. I would code it as equation written in Wikipedia. No hurry testing that, if this already runs for you. It would be more important to figure out the reason for the high posterior correlation. Or before that you could try with dense mass matrix, that helps with correlated posteriors, too (unless the number of parameters is very large and the multiplication with the big dense mass matrix starts to dominate the computation time)