I do not use the Rhasspy integration in HASS, I run the main node in a docker container. Rhasspy (main and satellite) is connected to the Mosquitto broker exposed by HAOS. The satellite pushes the STT, Intent Recognition and TTS via the Hermes MQTT handler to the main node over this connection. The acronym overload should become clearer when you start tweaking things in the Rhasspy web interface, it is very well done for the amount of customization you get.
The HASS side is done via yaml scripting. You can find my mpd.yaml at [2]. I select music to play via an ampd [3] interface hosted at the RPi connected to my HiFi (not the satellite, dedicated RPi). The sentences I use are at [4]. The names for HassTurnOff are the same as in HASS, this gets handled by built-in intents in HASS. I am considering trying out Node-RED, because I am a bit unhappy with the yaml.
I am also looking for a case. Currently I use [5] just hooked up to the ReSpeaker HAT 3.5mm and USB on the RPi with cables in a chaos on my desk. I also took apart [6] (smaller than you would expect, does not go back together nicely) and it might fit an RPi0W2 with HAT, if you take of its top. Both speakers are USB powered, so you can plug it directly into the RPi. I would prefer to stuff everything in the case of [5] and hook up the blue LED to a GPIO, but it is quite cramped in there, maybe it can "fit" by making a hole in the back.
[1] https://rhasspy.readthedocs.io/en/latest/tutorials/ [2] https://pastebin.com/M7KmQSiM [3] https://github.com/rain0r/ampd [4] https://pastebin.com/tsUqWwpy [5] https://www.amazon.de/gp/product/B07DDK3W5D [6] https://www.amazon.de/gp/product/B08GC8K8ZR