Do they work? From my understanding, older ones do not, newer ones do. How do I tell which ones do or not?
In particular, Pokemon Black and Chinatown Wars.
Pokemon Black is indeed a DSi-enhanced game. The page is correct (even though the Pokemon cartridge color isn't white like the picture depicts. I think the white is only for DSi-exclusive games, while some games are just DSi-enhanced, such as Pokemon Black/White.) The other game is just a DS game.
Unfortunately I don't know if there's a standard way to tell if the box doesn't mention DSi-specific features. Most DS games are not DSi-enhanced, your best bet may be searching for information on specific games.
As said earlier, your best bet is to search online if a game supports WPA2.
Unfortunately, most games don't and games that were released before the DSi will not be able to support it.
I don't think there are any DSi Exclusive Games released besides the ones on the DSiWare shop. There are some in Japan but none in America.
The game cards will have writing at the bottom that says either NTR or TWL followed by four more letters then, at least in North America I think, the last part will say USA. ex. DSi enhanced game: TWL-ABCD-USA (replace the TWL with NTR for non-enhanced)
As for the case itself, I think you can tell. On the back of my Pokemon conquest at the bottom right, next to the serial code, there is written TWL(Just like the letters on enhanced games) then some more letters ex. TWL P VPYT CAN(USA for the U.S. probably).
Anyways it's written going sideways right underneath the serial code if you hold the part you open upwards.
Hope this helps.