Can you copy the games to a flash cart as *.zip files and run them that way or do they have to be in the *.nds, *.gba, etc format? Just curious as this would save a lot of disk space for more games

depends on the flashcart and software you are using. some support compressed roms in zip format as far as i know.
eg. the g6 lite:
"Support decompressing ZIP format source file and compress game data in flashcard."
source: http://www.g6flash.com/feature.htm