You can vs your rival every monday on your game at the Pokemon League.( this also counts for HeartGold too.)
You can battle your rival at the Pokemon league on mondays and wednesdays
Yes.
no
You'll be introduced to him or her.
You cannot battle between a Pokemon Black game and a Pokemon Soulsilver game.
You actually get to name your rival. So, your rival is whoever you want him to be. The rival is just a strange character that happens to be eying Professor Elm's Pokemon.
lance
no. however, you do get to tag battle with your rival against Clair and lance after you beat the elite four in the dragon's den in blackthorn city.
About 10 times then you can fight your rival at certain days at the Pokemon League.
at indigo plauto
You only have one rival (or do you mean the second time you battle your rival?)
No.