If you cast a card "without paying its mana cost," you can't pay any alternative costs. You can pay additional costs, such as kicker costs. If the card has mandatory additional costs, you must pay those.

If there are no nonland cards left in that library, the opponent will exile all cards from their library, and Stolen Goods will finish resolving.

If you can't cast the nonland card (perhaps because there are no legal targets available) or if you choose not to cast it, it will remain exiled.

If the card has in its mana cost, you must choose 0 as its value.

You must follow the normal timing restrictions of the exiled card.

You cast the card from exile. Each land card exiled this way will remain exiled.