No.
Yes, the batter is credited with an at-bat and scored as a fielders choice.
Yes
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.
It does not count as an at bat
Nope. It's a stolen base.
No hit it is a fielders choice
No. This would be a fielders choice and would not count as a hit for the batter.
This is not a sacrifice. It is counted as an RBI though unless there is a double play as result of the FC.