The batter-runner is not forced-out at first base though a BR being put out before reaching 1B is treated similarly to a force-out in that no run can score if that was the third out.
Rule 5.08 (new system) 4.09 (old system)
It is not a DROPPED third-strike, it is an UN-CAUGHT third-strike. When you call it by its proper name it is easier to avoid confusion.
For the strike to be "CAUGHT" it must qualify under the definition of "A CATCH". From OBR Rule 2.00 - DEFINITIONS:
A CATCH is the act of a fielder in getting secure possession in his hand or glove of a ball in flight and firmly holding it; providing he does not use his cap, protector, pocket or any other part of his uniform in getting possession. It is not a catch, however, if simultaneously or immediately following his contact with the ball, he collides with a player, or with a wall, or if he falls down, and as a result of such collision or falling, drops the ball. It is not a catch if a fielder touches a fly ball which then hits a member of the offensive team or an umpire and then is caught by another defensive player. If the fielder has made the catch and drops the ball while in the act of making a throw following the catch, the ball shall be adjudged to have been caught. In establishing the validity of the catch, the fielder shall hold the ball long enough to prove that he has complete control of the ball and that his release of the ball is voluntary and intentional.
The key in the definition is "of a ball in flight". Also from Rule 2.00:
IN FLIGHT describes a batted, thrown, or pitched ball which has not yet touched the ground or some object other than a fielder.
If the pitch is not caught in flight on strike three then the batter is NOT out unless first-base is occupied with less than two outs.
Larry in TN's post in Does the run score? was marked as the answer