You choose which permanent the player gains control of, not the player who was dealt damage.

The player who was dealt damage can't refuse your generous gift. That would be rude.

If the target permanent you control is an illegal target by the time Blim's ability tries to resolve, the ability doesn't resolve. The other player doesn't gain control of your permanent, and no one loses life or discards cards.

Players lose life and discard cards for all permanents they control but don't own, not just the ones they received from you.

The control-change effect of Blim's ability lasts indefinitely. It doesn't wear off during the cleanup step, and it doesn't expire if Blim leaves the battlefield. In a multiplayer game, it does expire if the current controller leaves the game.