You decide whether to repeat the process after each iteration. In other words, you discard a card, see what it is, Kindle the Carnage deals damage, then you decide whether to continue. If so, you discard a card, see what it is, Kindle the Carnage deals damage, then you decide whether to continue, and so on.

Any damage prevention or regeneration abilities must be activated before Kindle the Carnage starts resolving in order to have any effect. You can't wait to see which (if any) cards are discarded.

A creature that's been dealt lethal damage by an iteration of this effect will remain on the battlefield and be dealt more damage by any further iterations. After you stop, all creatures that have been dealt lethal damage are put into their owners' graveyards at the same time.

All the damage has the same source. However, each iteration of this process is a separate instance of damage. An effect that prevents damage the "next time" this card would deal it will prevent damage from only one iteration.

A single regeneration shield is sufficient to save a particular creature, no matter how many instances of damage are on it.