You would make a second, duplicated scene with the overlay enabled and switch to that scene, then back. You could use the Video Player plugin to play a soundfile. To avoid having chat overlays reset, set them up as Global Sources. Likewise with the game capture and webcam and so on.
I do this for a 'cutscene' duplicate of my main-game scene, with chat and my webcam disabled, so as not to obstruct story elements.
To hotkey a single source, you will have to switch to OBS Studio. It doesn't allow sound hotkeying though, you could always get a soundboard app and set the hotkey to the same one as to toggle the source. Though that could lead to the sound playing twice. So it could be a good idea to have them on separate but adjacent keys.
If you're technically inclined you could possibly also set up a toast webpage overlay system and run it through the CLR browser, but that has its own challenges... including needing to know how to code it up, as there's no turnkey solution for that of which I'm aware. I don't believe that webpages run through CLR browser are handled interactively, either, but a custom Flash setup might be able to just grab raw key data and run off that. Again, gets into the technical side, nothing ready-made to do that.