two puzzling questions

[JEL]

#1, If offense brings to PU's attention, they are dumb! you must mean defense. If discovered before the next pitch, the batter who should have batted is out, and all advances are nullified. The next batter following incorrect batter is now up, that may be the one who just hit. If an out was made, it stands, and go to the next batter. Now if the offense did report the infraction, I tell coach I didn't hear that (this has happened) this is a defense appeal.
#2, This is confusing if you let it be. Rule is lengthy, but clear, POE states the same and does clarify. Your situation ...Girl can stop, run, wiggle or whatever until ball is in circle and pitcher has control. Look-back then comes into play, runner must go directly one way or the other, and if she stops again, she is called out. Note also, pitcher has no responsibility to "Look" runner back. I seem to get this question at most every game. The last was from 12U pitcher "I looked at her, and she was off 2nd" Yes , but she was going back thats OK.