If you're on hax 2.1, every homebrew application runs by default at 804MHz. If you're on hax 2.5, homebrews can switch back and forth between 268MHz and 804MHz when the code calls for it, but some will always stick with 804 regardless (like right now, for example, RetroArch).

If you're asking about CIA homebrews, it depends, some are always at 804, and some are always at 268 (but code for switching is still permitted).

 

Indeed. It's also one of the reasons the N3DS is able to run things like Nestopia a lot better than the O3DS.

 

Clock speed will probably be the big killer but I should say there were several instances of homebrew, and the occasional commercial game, on the GBA and DS having some suspect coding that might drain the battery faster than some properly optimised coding. I am not familiar with the low level aspects of the 3ds that might control this and the more modern nature of the processors (they are very similar to the ones that are in phones so it is not a big stretch of the imagination to imagine there being something more considered than the GBA stuff) but I would not be surprised to hear it was not used to the fullest by current libraries, compilers and homebrew devs.

Here is some of the stuff for the GBA
http://www.dwedit.org/dwedit_board/viewtopic.php?id=480