No, 'tear up' would be if she was pulling the pages out of the book. You actually used a more appropriate verb (to grab) before you tried to form the sentence, so it would be 'don't grab the book.' Although I've always been told when looking after my cousin that I should keep the message positive rather than negative, which means avoiding saying things like 'Don't do X...' But I digress.
We say 'turn the page' not 'flip.' It would also be an 'instruction', not an 'order.'
You can say 'sit on my knee' (the singular is much more common) or 'sit on my lap.'
Lap: "The front area from the waist to the knees of a seated person."
As for lay down vs lie down, someone else can tackle that one.