No, Pokemon Black and White are compatible with all versions of Nintendo's DS line.
The Nintendo 3DS is backwards compatible with Nintendo DS titles, so all DS Pokemon games and subsequent 3DS Pokemon games will work on the 3DS.
Yes , any game made for the DS will be compatible with the DSi and DSi XL consoles
No, Pokemon Black 2 and Pokemon White 2 will be available for both the Nintendo DS and Nintendo 3DS.
No, Pokemon Gold version will not work on the Nintendo DS. Pokemon Gold is a GameBoy Color game, and the Nintendo DS's GBA compatible slot only works with GBA (GameBoy Advanced) games. Even though the GBA system is compatible with Pokemon Gold, the GBA compatible slot in a Nintendo DS is not compatible with Pokemon Gold.
The 3DS can play all DS games, including Pokemon Black/White.
Yes, Pokemon Black and White can be played on any of the Nintendo DS family systems.
Nintendo DS Lite Handheld Console --- $197 Nintendo DSi --- $248 Nintendo 3DS --- $248 Pokemon Black or White --- $47 (big W) Pokemon Black or White --- $58 (Game) Pokemon Black or White --- $68 (EB Games)
Pokemon Black & White, Heart-Gold & Soul-Silver, Pearl & Diamond are all compatible with the Nintendo DS, DS Lite, DSi, and possibly for the DS XL and 3DS.
Pokemon Black and White will work on the 3DS like any other DS games and also supports the game's DSi-exclusive features. (This does, however, mean that region-locks apply when playing it on the 3DS)
When you say Internet, I think you mean Pokemon online, Pokemon indigo, etc. The answer is a no to that. These online games aren't approved by Nintendo and therefore you cannot transfer. On Pokemon.com, you can transfer Pokemon from the global link onto Pokemon Black or Pokemon White for DS
Yes you will be able to play Pokemon Black and White in the 3DS.