[MD Longhorn]

Sometimes an illegal pitch is delivered to the batter.

True.

If the batter hits the ball (fair or foul), the play continues until the "end of playing action".

False, it continues regardless of whether the batter hits the ball, in some cases.

Now the coach of the team at bat has the option of the resulting play or the penalty.

False.

The ball must be hit.

False.

If the batter swings and misses the ball, the play is dead, all runners return, and the award is made.

False, in one case.

If there is not any delivery of the ball to the batter, the play is dead and the award is made.

True!

The runner stealing has no bearing on the play.

I gave him this one. Technically not really true, but also not blatantly false. Just unimportant.