Hi and welcome!

I have a covered litter box with a flap and my cats still get some litter on the carpet when they're coming out after doing their business, unfortunately there isn't a whole lot that can be done. I have three cats though and have to vacuum every evening around the litter box. A litter box with a flap or curved edges definitely reduce the problem somewhat though. Putting a litter mat under the litter box might be a good investment!

As for the biting, thats a problem fairly common problem in kittens. If she bites you, don't pull away too quickly - it becomes a game of 'catch the finger'!
End playtime when she bites you and don't continue petting her and she'll soon get the message. I had the same problem with one of my kittens... I would get up and walk away every time he bit me. Don't take it personally! She's just playing! Tapping her on the nose has little or no effect on cats/kittens, or so I've found with my cats/kittens!

Hope this helps!!

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"There is, incidently, no way of talking about cats that enables one to come off as a sane person. - Dan Greenberg"