Streaming Music through my home stereo

The simple solution would be to stream wirelessly via Bluetooth from the iPad to a Bluetooth receiver plugged into the Pharaoh using a stereo mini to dual RCA cable. That limits the quality of any files that are of higher resolution than MP3 or Apple Music's proprietary AAC format but if Apple Music is your primary source the quality should be quite acceptable.

Here are a couple of well-reviewed options at different prices. The Audioengine B1 has the advantage of using Bluetooth with APTX which should make a small but noticeable difference.

https://www.logitech.com/en-us/product/bluetooth-audio-adapter

https://audioengineusa.com/shop/adapters/b1-bluetooth-music-receiver/

The most important thing by far with streaming is just to get the stream going. Anything beyond that is icing on the cake. You guys will understand when you get a little older. 

I don't know if you will ever be able to control it from your iPhone. But if you can get it to play a waterfall track that will probably help.

BluOS 

BluOS would certainly be a good choice for most music services but since Apple doesn't yet provide an API that would allow access by Blu it isn't an answer for someone who wants to use Apple Music. There are definitely other excellent choices of music services.

Why would you even want to use apple music? Isn't all their content compressed? I know it used to be.