I have count data in y variable and it is right skewed
For your first question: you no longer expect homoskedacity from a negative binomial model, but you do expect the outcomes, expressed as quantiles of the relevant negative binomial distributions, to be approximately uniformly distributed. And you still expect them to be independently distributed without severe autocorrelation. You might take a look at the posterior predictive checking options that
rstanarm provide for these models for some initial ideas about checks for misspecification.
Your second question (with mention of a Wald Chi-squared) suggests that you are probably not analyzing these models using Stan and are not interested in model selection procedures that are applicable in a Bayesian setting. We try to keep this forum focused on questions pertaining directly to Stan and immediately adjacent topics in bayesian inference. You might consider posting this question on Cross Validated: https://stats.stackexchange.com/