At the moment we do not have access to the DSi's upgraded hardware. Current DSi-compatible devices boot the system in DS mode, which is limited to the same hardware that the original DS/Lite had. We don't know if an exploit is coming soon or at all, at least not yet.
There is no GBA emulator for the DS or DSi because it simply doesn't have enough power to do it faster than perhaps 1 frame per second. However, on the DS and Lite, you can get a slot-2 device to run GBA games natively on the system at full speed. On the DSi of course this is impossible since it doesn't have a GBA slot.
SNES emulation is not going to be better than the PSP's. It will be faster in general, since the SNES emulators for the DS use hardware rendering rather than the software rendering used by the ones for the PSP, but they are graphically inaccurate and their compatibility is lacking. You'd be much better off figuring out how to get the emulator for your PSP running better (I know there are ways to get it to only need 1 frameskip for most games, but I don't know what those ways are).
Running NES games on a DS is simple - download NES DS, put it on your card along with your *.nes games, launch nesDS.nds, and choose the game to play. Same with Gameboy and Gameboy Color games, using the Lameboy emulator and *.gb/*.gbc files, and same for Genesis games using jEnesisDS. For GBA, assuming you have the slot-2 EZ-Flash 3in1 expansion pack (the most popular slot-2 addon device), you would launch the GBA ExpLoader software to load a GBA game from your slot-1 microSD card to the 3in1, then launch the game. For SNES, you'd download SNEmulDS, put "snemul.cfg" on the root of your card, put the games in a folder called "SNES" on the root of your card, launch SNEmulDS.nds, and choose the game to play.
So in short, NES and GB and GBC and Genesis games are all entirely playable. GBA games are playable if you get a DS Lite and a slot-2 device to run them. SNES is lacking.
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