"The ball is in your court" is a common phrase in British English. The meaning is that it is now *your* turn to act. For example, "we have tried to help her as much as we can, the ball is in her court now" (in this case, the other people have done as much as they can, the reasonability to act is with the other person).

The expression "the ball is in your court" is used fairly often in American English. It means it is your turn or responsibility to make a decision or perform an act.

That is correct. The bail's in your court is not a commonly used sentence in English. The ball's in your court is a metaphor (a thing regarded as representative or symbolic of something else), it means that you must make a decision in a situation.