Thank you so much for your incredibly thoughtful replies! You hit the nail on the head with the responses regarding modeling of SES. An additional consideration is multilevel modeling of SES, related to how you answered the second question. In addition, for modeling of effect modification, you can stratify or use interaction terms in regression models.
Your thinking about multilevel models for your outcomes are right on as well. Another way to think about your particular area of research is about the quality of care by hospital, with hospital being the contextual factor. Given how specialized this aspect of health care delivery is, that could certainly be a factor. Similarly, access to disability payments, paratransit, etc. could also play a role.
Hope you are doing well!