Your thinking about measures of SES on multiple levels is really important and I think would be interesting to explore in your work. There may also be policy level considerations (i.e. access to food stamps, etc.) that you could also consider as contextual factors.
For how to model SES, you can also think about confounding, although, as we talked about in class, this is complicated because the direction of the causal arrow from SES to race/ethnicity does not make sense - so it is really more of a "third variable" whose treatment depends on what your research question is (e.g. do you want to know what the race/ethnicity association is independent of certain SES variables, acknowledging they are usually imperfect measures, or do you want to know the association of an outcome with race/ethnicity without controlling for the mediating effect of SES).
Effect modification by SES is another consideration for modelling.
Thanks!