Q. I'm a little confused on castling. My brother says as long as the Rook and King are in their original starting places, they can castle. Is it possible that castling take place after one movement of Rook or King from their respective places?
A. No John, in order for castling to be possible, neither Rook nor King involved can have moved previously. It is also crucial that no other pieces are sitting between them. The King must not be in check and no square that the King must cross or arrive on can be under the control of the enemy. Here you can get further details on chess castling.
A. No Aswathy it is not permitted. Castling is not possible when the king is in check much less checkmate. Without exception, when checkmate occurs, the game ends instantly. The player whose king has been checkmated loses. Further reading available on castling should help you with this.
Q. Under what conditions may a player not castle his/her king for the time being?
A. You may not castle if your king is in check, or if either the king or the rook in question has previously moved. It is also illegal to castle if by doing so the king must cross a square under enemy control, or the king would be moving into check by castling. A more detailed discussion on castling is available.
Q. Can the elephant swap places with the queen on the home rank?
A. No this piece which is a bishop in Western Chess is not permitted to swap places with a queen. There is no provision for this maneuver between any pieces in this variant. The closest you could probably get to it is the castling maneuver between the king and rook.