Wow, not seen a R4 SDHC in a while (in case you were unaware the R4 name took off as name for flash carts and there were dozens of things with R4 in the name, often subtly different and mutually incompatible).
Not sure what ended up happening for that. If it is compatible with stock wood though then should be the last one which does pretty much every game just fine.
Consoles you can emulate.
There are three levels of emulation on the DS
1) Basic emulation available to everybody.
2) Extra RAM using emulators.
3) Enhanced flash cart (for most this means the DSTwo) with onboard processors that can do far more.
3) does have GBA options but that is not you here. Has quite a few nice things as well and things that do better than a stock DS (SNES is better, technically some minor PS1 stuff but nobody does that
2)
https://wiki.gbatemp.net/wiki/3_in_1_Expansion_Pack_for_EZ-Flash_V#Third-Party
1) ... lots. While there technically is a proof of concept (impressive effort, nobody is likely to use it for practical purposes) for the GBA most people will either need a GBA flash cart/expansion pack (at which point you are running GBA games in GBA hardware) or an enhanced flash cart (has some perks over hardware but is not quite as compatible). Mostly a minor perk for a few SNES games in one emulator and something for an old mac emulator.
Beyond that.
16 bit stuff gets a bit iffy. SNES and megadrive/genesis are a thing but the SNES will only play limited games and resolution issues with the megadrive (320x224/320x240 PAL vs the DS' 256x192) mean some oddities and scrolling screens there but actually you can play a lot. It lacks some of the more exotic stuff we saw on the GBA (many people had GBA flash carts throughout the DS so was easy enough there for them)
http://nintendo-ds.dcemu.co.uk/emulators-for-nintendo-ds-1158162.html is a reasonable start. Most of the older stuff works well, might not have as much polish as some of the GBA stuff ended up with but better than a lot of what has come since.
Plenty of stuff also got ported as homebrew, commercial games and more besides, as well as having ports on other systems that might be emulated (or emulated better).
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