Batting around the order happens when the 10th batter comes to the plate (actually the first batter up in the line-up)
If the batter touches home plate, while making contact with the plate, he is out.
Nothing is scored because of a batter stepping on home plate. There is no rule in MLB prohibiting a batter from stepping on home plate.
a home run
Absolutely he can cross the plate if he wants to. The reason they walk behind the umpire is just out of common courtesy. When you step over the plate, you risk the chance of kicking dirt on it. The batter can only switch boxes if he is a switch hitter and a new pitcher comes in. Otherwise a batter must remain on the same side of the plate for the entire At-Bat.
Many little leagues play a 9th batter rule. At the coach/machine pitch level our league plays that when the 9th batter comes to the plate, the hitting team must announce it or he's out after a pitch is thrown. Once announced, there are automatically two outs. That batter completes his at-bat, either safe or out and then the half-inning ends. For our Minors (kid-pitch) kids, same rules, except the 9th batter can stay on the bases and advance until he scores. When he crosses the plate or an out is made the inning ends. The only way runners behind him can score is on an over-the-fence home-run.
No. There is no rule prohibiting the batter from stepping on home plate.
A play in which the batter makes it safely around all bases and back to home plate without stopping.
Inside.
A play in which the batter makes it safely around all bases and back to home plate without stopping.
On the right side of the plate.
It's behind home plate
The batter is called a switch hitter.