Gfoley4

batter hits a line drive to the second baseman, he scoops it clean on a hop... is the batter out? Same rule applies, this is an uncaught third strike and the batter may try for first.

Any batter is out when: 
 

Disregarding a too quick "out" call, the fielder who made the tag must keep the ball in his glove securely. Unless the other fielder was able to tag the runner and also gain possession, sounds like safe to me.

Not a force out, this is an appeal and a time play. As long as the runner from third scores before the runner from second is put out, the run scores. 

That's a softball rule, I believe. 
PBUC manual page 84

A ball is not fair or fair until it bounds past a base or is touched by a fielder. If no fielders touched it and it bounced past first base in foul territory- not going over any part of the base- it's a foul ball.

You are correct in that there is no "automatic out" for multiple runners on the same base. 

Yes, this is an advantageous fourth out. From OBR, although it would be the same under high school rules. "Appeal plays may require an umpire to recognize an apparent “fourth out.” If the third out is made during a play in which an appeal play is sustained on another runner, the appeal play decision takes precedence in determining the out. If there is more than one appeal during a play that ends a half-inning, the defense may elect to take the out that gives it the advantage. For the purpose of this rule, the defensive team has “left the field” when the pitcher and all infielders have left fair territory on their way to the bench or clubhouse."
 
So the out on the runner from second would have considered to have never happened for scorekeeping purposes.

it's not a force but no runs can score when the B/R makes the third out before first base

Fair. Any part of the ball on line = fair