Old Gameboy games on the Nintendo DS via slot 2 flashcard

You can if you break off the two tabs that block them but they won't work (GB/C runs at 5 volts while a GBA works at 3,3 - and that's before the completely different rom headers and instruction sets)

You can play most GB/C games on a DS (using a GBA card in GBA mode with Goomba, or a DS card in DS mode with Lameboy), but:

1- it's of course software emulation (there's no GB/C processor in a DS or GB-Micro, and without the mechanical GBC/GBA switch in the card slot there would be no trivial way to activate it anyway)
2- (but you could call it a corollary of 1-) it won't allow you to backup or directly run your GB/C originals

 

Some GBA flash carts have inbuilt emulators to run GB/GBC games. Others don't and you will need to build a copy of the emulator with the ROM in it (GBA flash carts/code technically have versions of code to read SD cards and such but few pieces of homebrew take advantage of it), indeed you might even want to do that with the inbuilt ones as well as they are not always the best versions for a given task.

Simply trying to launch a GB/GBC rom in GBA mode will not get you very far.

 

You might want to just get a PSP, Vita, etc. for a better all-around emulation experience.