Yes.
Anytime a third strike is dropped the runner can advance to first unless he is tagged or the ball is thrown to first before he reaches.
yes
Yes, it is scored as an official time at bat and a strikeout.
Throw the ball back to the pitcher. A batter can not advance on a dropped third if first base is occupied.
Well, first of all, a dropped third strike is an out if the catcher's throw beats the runner. It's that way because that's just the rule.
no
No he does not. He can take the pitch and if the catcher drops it or the ball gets by him the dropped third strike rule is in affect unless there is somone on first with less than 2 outs. If there are 2 outs the rule is in affect even with a runner on 1st.
The "K" is placed backwards in cases where the batter strikes out looking.
The batter can make an attempt to reach first, but if the ball is fouled off by the batter, the batter is out (strike out).
On a dropped third strike, if there's a runner on first and less than 2 outs than the batter is automatically retired, whether or not the runner from first was stealing on the pitch. If there are 2 out, the batter can try to reach base, and the runner from first would be forced to try to advance to second. As on any other pitch, a runner can always try to advance, but would only be credited with a stolen base if he left the bag when the pitch was thrown, not after it was dropped.
It is scored as a strike out for the pitcher (as far as the pitcher's stats) but not an out against the team at bat. The base runner's advance to 1st is scored as a passed ball (error) on the cather.