Softball Rules: Understanding Tag Outs and Batter Rights After Dropped Third Strike

Question
1)  runner gets single and instead of turning to her right and staying in foul territory, turns LEFT and returns to 1st.  She is tagged out.  She had no INTENT on going to 2nd.  Is she still out because she is in fair territory or because she turned left.  Probably both right?
2)  batter swings through strike three, catcher drops ball.  Can she batter still run?

Thanks.
IP

Answer
For the runner to be tagged out in question 1, she needs to make an attempt to start to second base, otherwise, no out, no matter which way she turns.

With first base open with less than two outs yes.  With two outs, yes even if first base is occupied.

Tom