Yes, with two outs the uncaught third strike rule is not in effect.
No. There are 3 outs. It makes no difference if the catcher catches or drops the ball. A hitter may attempt to take first base if he strikes out and the catcher drops the ball AND first base is not occupied. In this case, first base is already occupied the inning is over.
If he is called out on strikes and the catcher drops the ball, the batter may try to run to first base. In order to be put out, the catcher must throw the ball and get him out at first before he gets to the base. If the runner beats the throw, then the runner is safe. The pitcher is given credit for a strikeout. The catcher is given an error. This is how pitchers can have 4 strikeouts in an inning, which happened in MLB this week.
This play is referred to as "redeye". If a batter misses (or does not swing) at the 3rd strike, and the catcher drops it, the runner must run to first before the catcher throws the dropped pitch to first. If the runner is beaten by the throw, it is simply a strikeout in the books. If the runner beats out the throw, it still goes as a strikeout, but his advance to first will be listed as an error by either the pitcher or the catcher (depending on how bad the pitch was, and the reason it was not caught). In Little League (60 foot basepaths) batter is out on strike three no matter what the catcher does.
yes, unless there are 2 out...then the batter must beat the throw to first for the run to count...just as if he put the ball in play, no runs can score if the batter is put out before reaching 1st base for the final out of the inning
In the event of a batter striking out but the Catcher dropping the ball and the batter advancing to first base, it will be ruled as a strike out however it will also be ruled as a passed ball for allowing the batter to reach base. Generally speaking, in most cases the catcher will throw the ball to first base before the batter gets there. In that situation, the put out is given to the first baseman & the catcher receives an assist.
The batter can advance because the ball is still live, same as every runner on any base. The catcher must catch the ball directly from the pitcher to complete a strikeout.
Yes, runner cannot intentionally dislodge ball from fielders glove.
Yes, if the catcher drops the ball
Yes the official scoring notation is K-E2 (strikeout - error on the catcher)