By rule, if a fair batted ball hits the batter while in the batter's box, it is a foul ball and a strike in FP if fewer than 2 strikes. What's considered in the "batter's box"? Do both feet have to be entirely in the box? Can any portion of one foot be outside the line and still be considered in the box?

Also, by rule the batter is out if they leave the box but then returns to the box and makes contact. Does the same definition for the previous question apply here? By leaving the box do we mean one foot completely out of the box? Both feet? Or any portion of one foot? When returning, does both feet have to be in the box?

Only 1 foot must be entirely out of the box. This could mean any portion of the foot that is touching the ground outside of the box (line is considered in the box). For example, if her pinky toe is touching the dirt outside the white line, and the rest of her foot is in the air, and she makes contact with the ball - she is out. As for your second question, I would assume that if she meets the above criteria, then returns to inside the box and makes contact, she is out as well.

Is she wearing toe shoes and a tutu? ;)

rwest to answer your questions see POE #7 and rule 7-6 D-F effect, that should clear it up.