Batters Box Question

[BK47]

I am a HS Baseball Ump, however have a couple of questions regarding some situations that happened in an Adult league softball game.

1. Are there batters box's in slow pitch softball? If so is it manditory for the batter to stand in the box while being pitched to? What, if any, penalty is there for not being in the box?

2. Can the batter, while being pitched to, run up past the plate, almost 2-3 feet past on some occasions, to swing at the pitch?

3. And whose responsibility is it to keep score? Team B thought, according to their scorebook, they were up by 2 runs while the field Umpire said they were down by 1 which cost them the game. Dont you go by the official scorebook, being each team has one, rather than going by what the Umpire says, especially what the Field Umpire says?