Framework

So you are talking about turning off the monitor with its own push-button?

This is complicated. HDMI and DP are designed differently. HDMI is far "dummer". So it really depends on how the monitor behaves when you turn it off (that can range from GPU cannot possibly tell its off to the monitor acts to the GPU like the HDMI cable was unplugged).

Since you have DP into your GPU, the USB-C hub needs to convert what it may be able to tell about the HDMI monitor into DP format.

I can tell you that default behavior for Windows is to switch to last active audio device and monitor configuration for the current set of attached monitors (I do not actually know what Windows ties audio devices to). So the first time you attach a new audio-sink, Windows will switch to that. If that was the active output when you disconnected it, it should switch to it, when you plug it back in. In fact, Intel and Nvidia drivers under Windows have a habit of declaring the same monitor as a new audio-device far too often, so it more often switches to that as output than it does not.