Evolution: Can the genotype frequencies change, but the allele frequencies remain constant?

Where do you get that populations aren't evolving?

A population might be in HWE for a particular trait if there is no selection on that trait, but that doesn't mean that there is no selection or drift happening on any trait!

Let me focus on the necessary and sufficient part: such a language is suitable in mathematics, but does not really have much meaning in evolutionary/statistical context. Instead we talk about rejecting the null hypothesis, which does not mean accepting the alternative hypothesis. Equally, a failure to reject the null hypothesis does not mean that this hypothesis is true.

Hardy-Weinberg is a null model of evolution with about a dozen of assumptions. It is not a model of "no evolution", but rather a model of how evolution happens all the time (but it seems like "no evolution" in the sense that not much interesting is happening). Rejecting this null model/hypothesis means that some of its assumptions are violated. However, if the genotype frequencies do satisfy HWE, it doesn't automatically mean that the assumptions of this model are satisfied.