Most of the time you do but it can depend on what plan you are on
There are only a tiny handful of 'DSi games', games which specifically need the DSi's camera to play. Most are even DSiware. So technically all DS game cards will work just fine on a DSi, there's no need to do anything to get them working.
If you want the same game progress, yeah. But you can just go to the wii channel on the wii u and you could play your wii games fine.
You need a PS2 to play the games not a PC
You Dont need a controller. I have p64 too and I`m playing Turok 3 just fine
No I don't think so I have terminator 3 Rise of the Machines for ps2 and the remote worked fine.
You need a console to play console games. Computers play computer games and a PS3 will play PS3 games or the Same for the Xbox 360 and the Nintendo Wii.
You do not need to have a Memory Stick inserted to play games if you have UMDs.
A proxy is not required to play games
Yes, regular DS games will play fine.
If you are going to just play the games with no intention of saving and game information, such as profiles, settings, etc., then you should be fine with an Xbox without a Hard drive. You might even be fine with a memory card for the Xbox. I personally like to play games and save profiles, settings, achievements and my advancement in the games so I need a hard drive. You can always buy the Xbox with no Hard drive and if you realize you need it later one, purchase it separately.
Yes, DS games are not region-locked. Any console will play any games.
An emulator