Hello I have a new 3DS (the small one) at firmware 10.7.XX and I wanted to now if there were any ways to get it to be able to run 3DS backups on it and Nintendo DS backups on it. Any information will be greatly appreciated especially if it doesn't involve buying another game to exploit it. If I have to I have no choice.
nope, you either have to buy an exploit game to downgrade to 9.2(
FOLLOW PLAILECT'S GUIDE) and install a9lh, or buy a sky3ds.
Via the a9lh way , you can install 3ds backups, known as .cias, and with sky3ds(please don't go with sky3ds, a9lh is better), you use .3ds roms, which get you an online ban in no time

For DS roms, if you want to actually
play them at a playable state, you'll need a DS flashcard such as R4i sdhc 2016 gold or R4i-Gold Card(DS mode cards)
If you want a buggy unplayable way to run DS games use
3DESMUME