Game Situations: Dropped Third Strikes

 
Runner on 2nd base (R2), no outs, and an 0-2 count on the batter (B1). B1 tips the next pitch. The tip hits the catcher's glove but then becomes lodged in the catcher's chest protector. Is this a foul tip and a live ball? Is this a foul ball? Should the batter and runner be awarded bases when the ball becomes lodged in the defensive's equipment?
 
Call
When a ball becomes lodged, it is an immediate dead ball... so in each of the three questions, the ball is dead, but there are different awards:
 
For the initial question, this started off appearing as a foul tip (the pitch tipped the bat and went sharply and directly to one of the catcher's hands). However, the ball then became lodged and uncaught, so we have a foul ball.

Bonus #1: The 0-2 pitch instead was in the dirt (for "ball-1") and then the ball bounced and lodged in the catcher's gear. What does the umpire do?
 
Call
For bonus #1, when the ball becomes lodged, the runner is awarded one base and the batter stays at the plate with a 1-2 count.

Bonus #2: The 0-2 pitch was in the dirt but B1 swung and missed the ball (for "strike-3") and then the ball bounced and lodged in the catcher's gear. What does the umpire do?
 
Call
For bonus #2, the batter struck out, but the catcher did not cleanly catch the pitch. With 1st base unoccupied -- the batter must be put out. Since the ball became lodged and is dead, B1 is awarded 1st base and R2 is awarded 3rd base.