No, the entire batters box is not in foul territory.
The little small triangle is in fair territory (if I understand correctly). More precisely, fair/foul territory is defined by fair/foul line and not affected by batter’s box.
If the ball settles in this area, it is fair.
If while starting to run to first he hits the ball, I believe he would be safe and the ball foul, as long as the umpire did not think it intentional. The rule is if “his fair ball touches him before touching a fielder” and I would think that if the ball settles on the ground, and he runs into it, this is a foul ball. But I could see others interpreting as interference.
