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In either case, it doesn't matter if those cards are still in that player's graveyard.

If you can't cast the discovered card (perhaps because there are no legal targets for the spell), you'll put it into your hand.

Abilities that begin with "At the beginning of your end step, if you descended this turn" will trigger only once during your end step, no matter how many times you descended this turn. However, if you haven't descended this turn as your end step begins, the ability won't trigger at all. It's not possible to put a permanent card into your graveyard during the end step in time to have the ability trigger.

You exile the cards face up. All players will be able to see them.

If you cast a spell "without paying its mana cost", you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the spell has any mandatory additional costs, you must pay those to cast it.

Multiple cards have abilities that begin with "At the beginning of your end step, if you descended this turn." These cards don't need to have been under your control at the time you descended. For example, if a permanent card is put into your graveyard during your first main phase and you cast Stalactite Stalker your second main phase, its ability will trigger at the beginning of your end step.