Dan's post "The Case for Flux" [0] discusses what kind of use cases would benefit from a Flux architecture. That still very much applies to using Redux as well, and is worth reading. The Redux FAQ also has a section on "When should I use Redux?" [1]
There's nothing _wrong_ with passing down values as props, even through multiple levels - it's just that most people dislike doing so. You could make the values available using React's "context" feature, although it's likely that wouldn't be the best choice here. There's also a variety of "not-quite-Redux" libraries out there that try to simplify the interactions with Redux ("no reducers", "no dispatching", "no actions", etc), and perhaps one of those might be useful.
[0] https://medium.com/swlh/the-case-for-flux-379b7d1982c6
[1] http://redux.js.org/docs/faq/General.html#general-when-to-us...