The DS doesn't have an operating system running in the background, so once you load a *.nds file, whatever was previously loaded gets shut out of memory. So no, you can't run a media player and a game simultaneously.
It's worth noting that DSLinux is capable of multitasking if you have slot-2 RAM available. In one virtual shell I can play back some mp3 files while I browse the web in another shell and type on IRC in a third. Of course the music gets choppy whenever I load a webpage since the DS isn't really powerful enough to handle it gracefully, but the fact remains that it's possible.
I've seen that video, but from what I understand, the hardware was not meant to do that and you can easily fry the speakers that way. Haven't tested it myself so I don't know for certain, but I'm probably not going to try it just for the sake of safety.
My speakers are just fine. After doing, maybe 7 tests of it. The speakers are fine. I don't understand how it can fry the speakers if you're just inputting sound from a source. That's like putting in a DS Headset with a Microphone, and blowing in the microphone and the DS reads it.
But overall, my DS Phat is dead due to some F2 fuse problems. Also some screen problems as well.