No..these are two different official scoring designations and have no relationship to each other. See Official Rules of Baseball on steals and fielding scoring methods.
No.
Yes, the batter is credited with an at-bat and scored as a fielders choice.
Yes
No matter what, in a fielders choice play the batter is given and 0 for 1 in the record books. It is based on the thought that if the defense had attempted a play on the batter he most surely would have been out. You misunderstood the question. If a fielder's choice is attmpted, but all runners are save and no error is scored it's a hit ... since the fielder's choice didn't actually happen.
Yes
1. Base Hit 2. BB 3. HBP 4. E 5. Fielders Choice 6. Dropped 3rd Strike 7. Catchers Interference 8. Runners
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.