If it's a cartridge game, you insert it into the slot at the top of the DSi. When you switch the DSi on, the game will be shown on the menu bar, select it or touch it with the stylus to play.
If it's something downloaded from the DSi store, it too will be on the menu, further to the right. Just select it in the same way as above.
Yes. All DS games work on all DS systems.
yes
All games that work on Nintendo DS will work on Nintendo DS-i
Yes.
Yes it can there aren't games that only work for dsi
Of course. All Nintendo DS games work on Nintendo DS Lites. some Nintendo DS/lite games do not work on the DSi, however (like guitar hero on tour) because the DSi has no GBA slot.
The company that manufactures DSi games is Nintendo. Nintendo is the sole owner of the Nintendo DSi console, and is the only company that distributes legal DSi games.
Yes, the Nintendo 3DS (as well as the 3DS XL and 2dS) has its own 3DS games that will not work for the Nintendo DS, Nintendo DS Lite, Nintendo DSi, and Nintendo DSi XL.
It downloads games at the Nintendo Dsi shop channel
Yes, since Nintendo DS games are not region locked, the Japanese DSi can play North American games. The Japanese DSi is not physically different than the North American DSi so accessories will work as well.
No Nintendo DS games work on Nintendo DS game consoles and maybe some other Nintendo and not on any of the Xbox or PlayStation game consoles regardless of the country or region.
Yes, all Nintendo ds games work with the dsi.