is it possible to set the steamdeck to “default” to always keep picking the steamdeck speaker as default audio out also when an HDMI is connected through the USB-C?

Some audio issues were introduced in the SteamOS 3.5 update (partly due to having to handle the OLED model around the same time) which causes the HDMI problem. Hopefully it will be fixed in SteamOS 3.6 or 3.7. I’ve found that Bazzite doesn’t have the issue, although obviously that’s an invasive change, and I understand it’s still a bit buggy with the OLED model.

how do y’all combine music and games?

I think doing what you want could be a bit technically involved. One way might be to have one device control the music, and then cast it to the deck with snapcast or similar. Then, if you can get a snapcast client on the deck to be persistently running in the background, any music that is played on the other device, will be heard on the Deck.

Or more simply, you could try pairing your Deck in bluetooth from another device, and then select that Deck as an output. This is assuming that the Deck allows this, and that your OS supports this (Pipewire on Linux definitely does).