The runner isn't awarded anything. If the fielder can get the ball in time, the runner can still be thrown out at second.
The runner is only awarded if the ball is thrown out of the playing field, such as the dugout or the stands. It is then declared a dead ball and is treated like a ground-rule double; The runner receives his extra base.
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When there are no outs the pitcher should always throw the ball to third base to get the lead runner. The third baseman now has the opportunity to throw to first for the double play as well.
If there is time you should always get the lead runner out. This keeps the force in play for the next batter.
Opinion/Answer:
This really depends on the level of Baseball you are playing and I guess to an extent what your coach wants you to do. However, in more advanced baseball (i.e professional or college), in this situation the pitcher should go to 2nd base and then to first for a 1-4-3 or 1-6-3 (depending on who covers 2nd on the throw) double play. In this situation you would have a runner on 3rd with 2 outs. In advanced levels of baseball this is an ideal position for the defense because you rely on the defense to get the out so the run wont score. If the pitcher goes to 3rd and then to 1st to try and get the double play you are 1. creating a longer throw from 3rd to 1st then you would have 2nd to 1st risking your chances at getting the double play. In most cases if you have a runner or 2nd or a runner on 3rd with 2 outs they will score on a base hit as they will run on contact --- so the pitcher should go to 2nd and then 1st for the easiest double play and leave the runner at 3rd (with the speed in professional baseball you are going to have a hard time getting a 1-5-3 double play, that being pitcher- to 3rd- to 1st)