Is there anything important to know (i know the were issues with hdmi cec at launch, for example), settings, Nintendo account?
Purchases are connected to the user (the Nintendo ID) that buys the game, but all users on the console can play the game. So if you buy things on the eShop, it would be best to purchase them on your own user profile, while your daughter plays on her user profile.
Is the switch app mandatory to play online with randoms on Mario kart?
Not at all. I don't think it's even possible to use the app when playing against randoms.
Yes.