Advancing on a Dropped Third Strike

 
One out and runners on 1st base (R1) and 2nd base (R2). With an 0-2 count, the batter (B1) swings and misses at the next pitch, but the pitch is dropped by the catcher. Seeing the dropped 3rd third strike, B1 begins to run toward 1st base. R1 and R2 see B1 running, so they also start to advance to their next base. The catcher decides to attempt to retire the lead runner, so he throws to 3rd base, where the base is tagged and then the ball is returned to the pitcher.
 
Who, if anyone, is out on this play? What are the umpire's actions on this play?

Call
On a 3rd strike with less than two outs and a runner on 1st base, the batter is out, even when the 3rd strike is not caught. Since the batter is out, the runners are not forced to advance. If the runners attempt to advance, they must be tagged to be retired.

So in the case below, the batter is out, but the runners have safely advanced. No action is needed by the umpires (except to make it clear to the batter that he is out as soon as he swung and missed the pitch).
Part 2:

In last week's situation, what if there are 2 outs to start the play? What are the umpire's actions?
 
Call
With two outs, when a third strike is not cleanly caught by the catcher, the batter is
not out. The batter must be put out by the defense. This situation creates a force -- both for the batter-runner and (in this case) for both his teammates (R1 and R2).

In the case below, the defense's actions forced out R2 at 3rd base -- so the inning is over!