Yes, if you are the batter.
It provides better visibility for the batter.
noWrong!! The batter may switch sides of the plate as long as the pitcher is not on the rubber prepared to pitch. He could switch several times during the at-bat.
I believe so. I think Shane Victorino does it.
if the batter swings and misses or if the pitch is between the chest and kness
Yes they can no matter the pitch count.
If it hit the batter the batter gets first and it's scored as a hit by pitch.
You can move to the other side of the plate during an at bat IF you ask for time out from the umpire AND time out is granted.
In softball yes.. if first base is not occupied.... but it depends on baseball and softball and what leval you play at