They are activated within the Unova Link menu and will run on the save file until deactivated.
No it doesn't delete your save file. If you use Challenge Mode on yourself the only thing it will effect are the Pokemon Breeders that are rechallengable, the Elite 4 and Champion, and any Trainers or key characters you have not beaten yet. Everyone you are not able to battle will not be able to battle you even with Challenge mode activated so you can't rechallenge the Gyms.
You can however increase the difficulty of another game you own if you wish to play through on a harder difficulty.
I don't know if Challenge/Assist Mode effect World tournament though as those AI's would already be at their highest point I believe.