Here is the situation. Runners at first and second nobody out. Grounder to second throws out the runner going to second. Called out clearly by the umpire. The relay to first for the double play is not in time. The runner to second who was called out continues his run towards third base dugout....however the first baseman tries to throw out the runner on third who was off the bag a fair bit but the ball gets past the third baseman. The runner on third will score but then the out runner who was jogging towards home decides to try and deke out the third baseman by touching the bag and breaking for home. It works. The third baseman throws home and throws that away allowing the runner on first to advance. What's the right call???
The runner that has been put out can continue to run the bases as long as he does not interfere in the play. It is up to the defense to know that he is out and forget about him. As rule MLB Rule 7.09 d) states,
As Paul states, in MLB Rule 5 the umpire will put the ball into play and stop the play. However with runners on base and continuing to advance there would be no situation where the umpire would stop the play. The only way the play would be stopped is if the defense was in complete control of the baseball and the runners on base. Meaning the runners are not advancing or trying to advance. Usually in this situation the defense would then ask for time to be called unless the umpire calls time to change balls, clean home plate, etc. If the defense does not call time, the play may remain live through the next pitch/play.
This situation is the same for high school (NFHS) and NCAA. I am not sure how it would apply to lower levels. It is a classic example of the defense throwing the ball around and not being aware of the situation. The offense should not and will not be penalized for that (under correct enforcement of the rules).
If the player that is put out at second base interferes with the play according to MLB Rule 7 then the play will be stopped and the correct punishment will be enforced.
Also note that if the player is put out, continues to the next base and the defense tries to put him out again, if the player is struck with a throw it is not interference so long as he does not make an attempt to get hit. It is the defensive responsibility to leave that runner alone
That could possibly be considered interference. The initial throw to the third baseman was legal and just a bad throw, with a legal run scoring. On the second throw to home, however, play should have been stopped by the umpire. Once a player is called out they are to return to the dugout directly, but his continuation on the base path was not necessarily an interference (stated below). This means that play should have been stopped and the batter should have been at first. Section 5.02 reads:
"5.02 After the umpire calls “Play” the ball is alive and in play and remains alive and in play until for legal cause, or at the umpire’s call of “Time” suspending play, the ball becomes dead. While the ball is dead no player may be put out, no bases may be run and no runs may be scored, except that runners may advance one or more bases as the result of acts which occurred while the ball was alive (such as, but not limited to a balk, an over- throw, interference, or a home run or other fair ball hit out of the playing field)."
Although, section 7.09 Paragraph e states: (e) Any batter or runner who has just been put out, or any runner who has just scored, hinders or impedes any following play being made on a runner. Such runner shall be declared out for the interference of his teammate;
In the situation you describe, according to the MLB Official rules, the runner cannot be considered an interference on that alone. It does, however, fall to the responsibility of the Umpire to determine when play should be stopped for any reason.