DS and DS lite have a GBA slot and thus can get a GBA flash cart to play GBA games natively (
https://gbatemp.net/threads/buying-a-gba-flash-cart-in-2013.341203/page-18#post-4756995 ). There is a proof of concept GBA emulator for normal DS flash carts (impressive piece of code but nothing you will want to use for fun) and also the option for enhanced flash carts (mostly meaning the Supercard DSTwo, which is rare as rocking horse shit these days, which will do about 95% of the library just fine in addition to a whole bunch of nicer emulators than the baseline DS can manage).
The DSi (and 3ds) have additional security that blocked most older flash carts, and eventually a lot of newer ones as well (though there are many options post that and options to revert to older).
Since then the DSi (and 3ds) have seen custom firmwares developed. GBA emulation is also a thing on them and not bad.
https://dsi.cfw.guide/ is a start, not sure what the current hotness is there for guides to that one. Better homebrew emulators are also a thing in this, even more so if you go for the 3ds.
DSi capable flash carts consist of mostly the few that survived the purges (or you revert to older/lose the blocks with custom firmware) that are probably quite good, and modern R4 clones for which you will want to learn about time bombs ($20 flash cart is nice, $20 year after year as they "break" is better if you are a seller of such things) and pick accordingly. The latter clones will probably lack fancy features like savestates, in game guides... but should do games, cheats, reset, homebrew and the like just fine which is all most really want.
DSi/3ds capable flash carts tend to operate in DS mode so no DSi extras for those few games that have them, no DSi downloadable games and no DSi homebrew (various DS homebrew got a DSi version that made use of the extra power) unless you put a custom firmware on it.
There is a vanishingly rare DSi mode flash cart (see cycloDSi) but don't even bother.
There are no region locks on the DS aside from the Chinese model (which has long since been bypassed). DSi mode stuff can have a bit in DSi mode but not for original DS.
There is some scope for GBA slots to be used to improve DS mode homebrew as well (
https://wiki.gbatemp.net/wiki/3_in_1_Expansion_Pack_for_EZ-Flash_V#Third-Party plus a more recent version of command and conquer) but DSi mode should obviate most of that.
On Japanese DS models then yeah they are cheaper most of the time and probably better quality (third rate Japanese might not be quite flawless outside it but close for many purposes). That said I still see plenty at yard sales, flea markets, car boots and the like so play that as you will.