Yes, but it has a different name its called dsibrew and it is extremely hard to download.
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The Mii channel is not available for the Nintendo DSi.
Launches apps not made by Nintendo
It isn't bad for you. but it will brick your Wii if Nintendo finds out you have it.
No, you cannot, because the homebrew channel is not official software. It is freeware and thus cannot and will never be able to be bought through Nintendo because it is free. Also, be wary of the homebrew channel it can and sometimes will completely crash your Wii (equivalent to the blue screen of death for a computer).
In America: A Nintendo DSi is $140; a DSi XL is $160. In Japan: A Nintendo DSi is ¥15,000; a DSi LL is ¥18,000. In the UK: A Nintendo DSi is £150. In Australia: A Nintendo DSi is $300.