According to the Official Nintendo FAQ it is 65 feet maximum:
Nintendo DSi - PictoChat Troubleshooting Because PictoChat uses the wireless feature of the Nintendo DSi, the following general troubleshooting steps may solve any problems that you are having.
* Make sure you are near enough to the others you are trying to chat with. The range of any model of Nintendo DS is up to 65 feet. * Check for any other devices nearby that could possibly cause interference, such as wireless LAN, microwave ovens, cordless phones, computers, etc. * Make sure there are no obstructions between the systems, and that the systems are facing each other. * If you are experiencing difficulty with the accuracy of the stylus and touch screen, try recalibrating the screen. For help recalibrating the touch screen, click here.
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You can pictochat up to 65 feet away from one another. It goes this far on DS, DS Lite, and DSI. Not 3DS. Wich sucks. Ihave a 3DS and cant pictochat,but Can communicate with SWAPNOTE!
So far Moving Notepad is only out in Japanese version of DSI.
lenght:9inches Height:about 2 inches how far back it goes: 5 inches
All versions of the Nintendo DS use wi-fi to connect to other DS's in range. As with most other wi-fi devices, the range is around up to 50 meters, but it can work further depending on interference and if anything is in the way.
As far as I know, Nintendo DS does not have sleep mode.