No the run does not count because the runner crossed home plate after the third out was made.
Another Answer:I read the question differently: "...runner on 1st is stealing and gets tagged out but after [the] runner from 3rd scores..."
If the runner from 3rd scores before the runner from 1st is tagged out, the run counts.
Yes, the run counts because there are only two outs.
The runner is allowed to leave as soon as the ball touches the glove as that is the beginning of the catch.
The runner is out and the batter gets a new at bat when the next inning starts.
The runner would be out if he does not return to the base before the fielder throws it to the base. Runners may advance from their base as soon as the ball is touched by a fielder. In this case, the runner would not be out.
yes
Yes, courtesy runners are put in strategically for stealing bases, or having a better chance to score on any hit.
No.
The plural form is runners up.
yes a sacrifice fly must be caught in fair territory. if it is not in fair territory when caught runners can not advance. and the base runner must also tag up after the ball is caught
The runner is out.
No. On any foul ball that is not caught, the ball is dead and runners may not advance.
If the ball is caught in foul territory, then the runner has the ability to run at their won risk. But if the player drops the ball in foul territory, then the runners have to stay at their respective bases.