The same player chooses both times. In a multiplayer game, you can’t choose a different opponent to repeat the process.

The player can’t choose to discard two cards unless they have at least two cards in hand. Similarly, that player can’t choose to sacrifice a creature or planeswalker unless they control one. This means that if the target opponent doesn’t have two cards in hand and doesn’t control a creature or planeswalker, that player will lose 5 life and then lose 5 life again.

No player may take any actions between the two times the player goes through the process.

Any abilities that trigger while Remorseless Punishment is resolving will be put on the stack after it has completely resolved.