Can a batter leave the bat on his shoulder when in the batters box?

Answer 1

Yes, batters can be ejected while they are batting, usually by arguing about the strike the zone with the umpire. This usually only happens after the batter has struck out, rather then in the middle of the count, but if the batter argues too much, the umpire can toss him at any time.

Answer 2

Absolutely. It's probably not the best posture to have the bat resting on your shoulder, because it takes away from the momentum of swinging the bat when you have to raise it off of your shoulder first, but there's no rule stating how the batter has to stand, as long as he's inside the batter's box. Some batters hold the bat way above their heads, while others have held the bat out in front of them. Chuck Knobloch (spelling) held his bat in front of him with the fat end of the bat facing straight back towards the backstop. Some batters have practically stood facing the pitcher's mound.