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In redux terms, your controller would be the bundle of "action creators". If you use vanilla redux, those would conduct all your "logic", including communicating with the backend. This has some pros and cons, so there are extensions("middlewares") to redux which iterate on the concept - redux-thunk, redux-saga, etc.

Since redux largely defines the design of the "Model" part and React largely defines the design of the "View" part, most of the variation when using React and redux is how to design your "Controller".

It is indeed very close to how you would model redux apps. Interesting that you mention that your controllers don't have read access to store(s). That is actually very common with redux-thunk or redux-saga.

AFAIK it doesn't, but I'm just learning. My understanding is the standard way of doing things work if you are creating the most boring CRUD apps but you need to "bend the rules" a bit to make it work for your use case very often - as long as you keep it newcomer-friendly.