The runner is out unless they are on the bag in time .
If the runner is in foul territory, it is just counted as foul.
in the MLB you can run to first if you swing on a third strike and the catcher does not catch it, he has to tag you or throw down to first, it's rare that the runner is safe
Assuming there is no force at home, chase the runner towards third base and that increases your chance of getting one or perhaps two outs. By throwing home, again assuming no force, the runner could get back to second safely and the throw to home is meaningless.
This is Interference, but whether the runner is called out depends on what happens. A throw to 1st base where the runner interferes with the play is certainly grounds for them to be called out. Also if the throw to 1st base hits the runner in fair territory, they should be called out for interference. The fielder should throw to 1st base as if the runner were following the rules and running in foul territory. If they choose not to and move to allow the throw to reach 1st base without hitting the runner reaches safely, the umpires are allowed to interpret the rules as though the runner is either out or safe.
Yes, the run counts. A dropped 3rd strike with 2 outs is not an automatic out and all base runners are allowed to advance, including the batter that just became a runner. If the throw down to 1st is not a tag but a force out then the run would not count. However, since it is a tag, the play becomes a timing play. If the run on 3rd touches home plate prior to the tag on the batter runner then the run will count.
the third party dies down
it will throw up
The world will end
Possibly it landed on the roof.
it can speed up and slow down depends how high you throw it and how hard and far.
Yes, the Official Scorer will credit a batter with a RBI in the event of a runner scoring from third in the event of a player getting caught in a run down however Official Scorers also will have the right to not credit the batter with a RBI in the event that they do not believe that the batter would have advanced the runner on third base to home plate.
They will roll towards the far end of the lane.