I had the same problem, and formatting did not work.
But to answer your question, formatting wipes your settings so it's like a brand new 3DS system. You'd have a new friendcode, mii plaza is cleared, everything.
eShop data is kept separate though, so anything you've bought from the store will still be there, you'd just have to re-download it all.
Mario Kart 7 will be fine since it is in the cart itself, not the 3DS. Gamesaves for eShop titles will be wiped though.
If I were you I'd call up Nintendo, use the warranty on the thing without formatting. (I say call because on the repair section of the site nothing really matches the description for DS mode wireless not working) There's a possibility that you might get to keep your friend code / system settings and even get a new 3DS.