Yes all the buttons across the bottom work. I don't have any TV out cables so I can't test any of that and the manual is in Chinese unfortunately so I can't read it. It's about the size of a PSP, but the only thing I have to compare it with is my DS lite.
Ah, functional buttons means it would have different emulators. An older unit like the A330 has to use key combo shortcuts for those functions. So even if the supplied emulators were better, the hardware compatible, and the OS unchanged we wouldn't be able to access the menu... Unless they didn't remove the old shortcuts. Can you access emulator menus by pressing START+SELECT still?
They also probably didn't give TV-out cables because of the confusion the last ones generated. The TV-out jack didn't support audio, which confused the heck out of users since the emulators used only one of the three supplied prongs to connect to the TV. The other two prongs were used for video playback, but people were trying to get sound out of them.
...and of course you only have a DS Lite, it's the only unit I don't have on-hand to look at.