yes it's an at bat
It does not count as an at bat
A fielder's choice is just like an out and lower's a player's batting average since the fielder chose to put out another player instead, usually the lead runner... On the other hand, a sacrifice does not count against the batter because his intention was to advance the runner at his own expense.
No.
Yes, the batter is credited with an at-bat and scored as a fielders choice.
Yes
I AM A COLLEGE BASEBALL PLAYER ALL INFORMATION HERE IS 100% TRUE. A fielders choice does not count as a hit for the batter even though the batter reaches base safely. It counts as an out for the hitters batting average but RBI's, runs scored, stolen bases and any other effects of the batter reaching base safely count. In essence a fielders choice counts as an out because the fielder could have chose to get the batter out or another runner which means there was an out on the play. Also, you cannot assume a double play even if the ball hit was the easiest 4-6-3 which the fielders made an error on.
Yes it lowers it. A fielder's choice affects you the same way an out would. It would be recorded as 0 for 1, FC.
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.
No. No error can ever be given when there is an out made on the play.
Nope. It's a stolen base.