Situation: Bases Loaded, 2 outs, 2 strikes on the batter.
Next pitch a called third strike is dropped by the catcher.
Question: Can the batter attempt to reach first and the all runners advance or is it the third out since 1st base is occupied?

The batter may try for first. Also, whether he runs or not, he has become a batter-runner, which forces everyone else to run as well. The cacher, in control of the ball, may simply tag the plate for the third out. (Assuming he does so before R3 touches home.)

Dropped third is in place when 1) there are two outs, OR 2) there are less than two outs and first base unoccupied at time of pitch.

It's easy to remember this when you think of not providing an advantage to the defense. For example, it there was 1 out with bases loaded, the catcher would always drop the third strike, then get an easy double play by stepping on home and throwing to first. So rule in place to avoid this advantage.