At the moment the work on the Rust API is on hold, probably indefinitely, whilst we work out exactly where we want to head with it. There is no documentation because it is not a released product, and is not in a position to be released. The main takeaway is that there is only one member on the team that actually knows Rust, and unfortunately due to increasingly busy real-life schedules the rest of us are not in a position to learn it in order to support it. So if something was to happen to that one developer, we're screwed! See this wiki on the "Bus Factor".
We are also exploring the possibility of adding an HTTP API directly into FTL, but we have also have some stuff we want to get out before that.
As with anything ever, there are no ETAs, but we will communicate as and when we are due to/have released something.