yes but it is homebrew
nope
yes it is called DScraft
Yes It works on the DS Lite, DSi, and 3DS.
DScraft is a DS homebrew adaptation of the popular game Minecraft. At the moment this is a rather complete adptation of "creative mode" (or minecraft Classic), which means there is only exploration and creation in this version. I am planning on attempting to adapt survival in future versions !
For the Nintendo Ds Lite, Minecraft is only available as homebrew meaning a homemade game on either an emulator or a Storage Device such as an R4i. If one has either of these two things, they just need to search for DScraft which is a small copy of Minecraft.
No, but if you've hacked your DS, I've heard that a game similar to Minecraft but with much less blocks and items, DSCraft, is available for you to download. Just search for it and you should find it.