Let's take a look at a situation where a third strike is dropped by the catcher.
The Defense--Back That Bag Up
The ball often goes to the backstop in this scenario.
The 2nd baseman should absolutely back up that throw from the catcher. It's a long throw, it's rushed, and not typically practiced. Note: The rightfielder should also sprint to the line to back up that throw.
The Offense--Make The Most Of It
The baserunner can only run to first base if it's 1. unoccupied or 2. there are 2 outs. This is a rule that is often misunderstood.
The baserunner must stay within the running lane or could be called for interference if he obstructs the catcher's throw. Nice fields have this lane marked. Most fields don't. It's up to the umpire's discretion.
As a baserunner, there is nothing wrong with running to 1st when it is occupied in this situation. Often times it will cause confusion for the defense.
What is the highest # of strikeouts a pitcher can have in an inning?