Prior work has uncovered a set of common problems in state-of-the-art context-based question answering (QA) systems: a lack of attention to the context when the latter conflicts with a model's parametric knowledge, little robustness to noise, and a lack of consistency with their answers. However, most prior work focus on one or two of those problems in isolation, which makes it difficult to see trends across them. We aim to close this gap, by first outlining a set of -- previously discussed as well as novel -- desiderata for QA models. We then survey relevant analysis and methods papers to provide an overview of the state of the field. The second part of our work presents experiments where we evaluate 15 QA systems on 5 datasets according to all desiderata at once. We find many novel trends, including (1) systems that are less susceptible to noise are not necessarily more consistent with their answers when given irrelevant context; (2) most systems that are more susceptible to noise are more likely to correctly answer according to a context that conflicts with their parametric knowledge; and (3) the combination of conflicting knowledge and noise can reduce system performance by up to 96%. As such, our desiderata help increase our understanding of how these models work and reveal potential avenues for improvements.