this gets to be question it every level.  I coached 12 and under team this summer and noticed this to be approved reoccurring problem.  let's go over the rule again about a drop 3rd strike and a runner advancing to 1st.

If a catcher does not catch a pitch directly into his mitt (it's not clean if it bounces or comes off of his equipment) then the batter runner may try to advance to 1st base if 1st base is not occupied and are less than 2 outs.  This is where the confusion always comes it with 1st taste occupied some runners will try to take off out of habit since the defenses responsibility to know that he cannot run the umpire will call them out but if you throw to 1st you're doing so at your own risk for no reason.  If 1st base unoccupied then you can run regardless of the outs.  Now the one that causes so much problems especially among parents by the way, is with 1st base occupied and 2 outs.  If this situation is true then the runner  made try to advance to 1st base.  We need to keep in mind that the force rules all apply here.  The catcher could throw to 2nd, 3rd, or touch home plate depending on how many people were on base. You do not have to throw to 1st base  you can text the batter runner and taken out, you can text the base in advance for the force out is always to take care of that dropped 3rd strike.  Beekeeping in mind with less than 2 outs and for space occupied  the batter runner cannot run and therefore should not be any throws involved.  But with 2 outs the batter may run regardless of for space is occupied or not.  This occurs at every level was even watching professional game of the night  between the Rockies  and Oakland Athletics..  There was a runner on 1st base with 2 outs, the count happened to be 2 strikes and 3 balls on the batter when the pitch was dropped.  The batter runner to coffer 1st base but the catcher apparently had a mind lockup because he popped up and through the 2nd like he was trying to throw out  the runner stealing was advancing because of the 2 out and 3 strike 2 ball count on the batter.  Course no one was covering in the throw went all the way to centerfield.  Luckily they got the batter out the didn't cause anybody but again remember the rule and the fact that even the best get mixed up at times.