"So, if I am thinking right, the Teac is on the .ext network by default of the ethernet cable plugged into the extender." Correct.
"Connecting the tablet also to the .ext network will then put it on the "same" network." Yes, and that is necessary for it to work.
"So it controls the Teac through the extender?" It controls the Teac wirelessley via the extended network, yes, by sending control directions, if you will.
"Meanwhile, the the music comes wirelessly, to the extender and on to the Teac? Is that the path?" Well, first of all, the music does not come wirelessly from the tablet. The tablet only provides control functions, it doesn’t play into providing the music (unless you were using Bluetooth to send content (e.g., a YouTube video) from the tablet to the Teac but that is a separate discussion not relevant here as I understand it). So the music comes either from a streaming service (Tidal/Qobuz/Spotify etc) to your router and then wirelessly to your extender and then though the cable to the Teac, or from your stored music collection located on a hard drive somewhere in your house and via your network(s) in some fashion (combination of wired and wireless likely, eventually getting to your extender and through the ethernet cable to the Teac), but not through the tablet (tablet just for controlling functions).
That is my understanding anyway - if I’m wrong about any of that someone will correct hopefully