It may be possible to run the games natively somehow due to the fact that the DS uses an ARM processor. Emulation I doubt would work, the DS sports dual arm processors one running at 33Mhz and the other running at 67Mhz. Emulating stuff is not verry efficient, but due to the simular processors someone may be able to make a library to make the games run natively. I have heard that the carts are just some modified type of SD card, but I don't know if that is true or not. There are usb SNES and Sega Genesis/Mega Drive cart slots so I don't see why someone couldn't make one for the DS other than legalities.
At any rate you ares still talking a lot of work, I have no idea if anyone would be interested in it or not. Furthermore, I can gaurantee that with the DS still being a cash cow for Nintendo if the original carts werent used they would sue. Even with the original carts they may claim the user in violation of the terms of service, and sue.
I have no idea what emulator works best with ds games since I just play them on original hardware. The other point you may consider is how you intend on playing games that require touch screen input or mic input. This would be far from ideal in either case.