A fielder's choice is when a fielder has choosen to make an out on another runner on base when he could of gotten the batter/runner out instead. The batter/runner is safe, therefore reaching base via a fielder's choice. Please note that this counts as an at-bat and goes against the batter's batting average.
A fielders choice is noted with the abbreviation FC and an indication of the path the Baseball traveled. For example...
With a runner on 1st base a ground ball is hit to the third baseman His throw to second base results in an out but the runner beats the relay throw to first. The scoring would be "FC 6-4"
No.
Yes, the batter is credited with an at-bat and scored as a fielders choice.
Yes
Earned. Unearned runs only apply to cases where an error allows a runner to score where they otherwise would not have.
Yes
Anytime the batter is put out at 1st base on a batted ball it is not considered a fielders choice, it is simply a put out. Now in your scenario if the catcher would have thrown home and the runner coming from 3rd is out and the batter is safe at 1st, then in that case it would be a fielders choice
It can't ... a balk can only be committed by a pitcher.
I assume you mean if you're score keeping and not saying "How would the batter who becomes a runner be able to score a run on the play." If you're referring to score keeping I believe you would just note that the batter reach first base as a result of the fielders choice by 3Bman. FC - 5.
No. No error can ever be given when there is an out made on the play.
It does not count as an at bat
Yes, the batter will usually be given an RBI on a Fielder's Choice in the event of a run scoring the only way the home team's official scorer will not give him an RBI on a Fielder's Choice is in event of the runner in the scorer's viewer not originally being able to score on a player and was only able to do so as a result of the defensive team's miscue.
Yes. If in the Official Scorer's opinion, the batter would have been out at first base had the fielder chosen to make that play then the play is scored as a fielder's choice. The batter reaches first base safely, but is not credited with a hit and is charged with an at-bat. He's officially 0 for 1 on that play.