The Elm and Reagent (Clojurescript) communities build apps this way. It's theoretically possible to do everything in render in practice you wind up wanting to do things in other lifecycle events (e.g. display a google map) in most apps. There isn't any component local state but the components aren't pure render functions either.

That should be possible! Of course, stateless components don't have state, so you would need to something like Recompose if you need that. And of course, not all React-related projects explicitly support stateless components at the moment (e.g. hot reloading).

Another option is to use psuedo-local state with Redux and pass everything as props: https://github.com/rackt/redux/issues/159. As Dan notes in that issue though, it might be better to wait for the state tree model to be improved: https://github.com/facebook/react/issues/4595.

In short: early days, too early to tell what patterns will emerge.

I generally try to do everything in render(). It almost always works out better. Sometimes I end up needing a lifecycle event. Timers (e.g., for manual animations) end up being hard to model without mount/unmount.