Yes, PocketNES works just fine on the 3-in-1. It doesn't search for roms. Like all emulators for the GBA, the roms have to be packed into the .gba file with the emulator. You should be able to get the app to do that from the same place you get the emulator.
PocketNES saves to SRAM, and the saves are managed just like the rest of the GBA saves.
Someone over on the PocketHeaven boards said that the only part of Earthbound Zero that doesn't work on nesDS is starting a new game. If you start a game on a different emulator, save, then transfer the save over to nesDS, the game supposedly works just fine. It will probably be easier to do this with a PC emulator than PocketNES.