The Wii U is a different console to the Wii. The Wii U however is compatible with Wii games and accessories.
No, the Nintendo Wii cannot play Wii U games
Yes, the Wii U console can play Wii U games
Yes, the Wii U is fully backward compatible with Wii games, Wii downloads, purchased WiiWare and Virtual Console games and has the Wii Menu built into it. However, Wii games cannot be controlled with the Wii U GamePad and will not be displayed in HD on the Wii U console. Also, GameCube Contollers will not work on the Wii U even if the Wii game allows the use of GameCube Contollers. There are GameCube Controller Adapters (Mayflash) that were created that the GameCube controller to function like the Wii Classic Controller, but is not an official product of Nintendo.
No, the Wii cannot play Wii U games. It's actually the opposite - the Wii U can play Wii games.
Yes, the Wii U is backwards compatible with Wii games. You need to insert a Wii disk, then go into Wii Mode by selecting the option from the Wii U menu. You'll need a Wii controller to use Wii Mode.
No, the Wii U cannot play Nintendo DS games
Wii U games cannot be played on the Wii.
No, you would need a Wii U and a Wii U game Pad to play Wii U games.
No, you would need a Wii U and a Wii U game Pad to play Wii U games.
The Wii U's gamepad is not a standalone console like a 3DS, it is in communication with the Wii U's main console body itself which is the thing running the game, and can't do anything without it.
No. Black Wiis are the exact same as white ones. Wii U games can only be played on Wii U consoles exclusively.