Sorry to just now be getting to this - and thank you for your thorough and thoughtful answers! In your answer regarding SES modeling you lump together controlling as a confounder and looking for interaction/effect modification, so it is important to be clear that those are two different modeling approaches.
If I understand your question about the ecological fallacy correctly, the point they are making is that both the ecological approach and the individualist approach are sub-optimal in isolation, and that be doing a mixed effects approach you can take both contextual and individual factors into account.
Your study sounds fascinating (and complex). For the question about randomization, I am assuming the randomization was stratified by geography? In general, in RCTs, you do NOT need to control for anything, and your primary analysis should be unadjusted (except for adjusted for clustering). You can always do a secondary analysis controlling for imbalances if your descriptive analysis shows that this might be useful.