Deep goblins laugh at humans' wands. They cast spells with their hammers.

Ore-Rich Stalactite // Cosmium Catalyst

{1}{R}, {T}: Choose an exiled card used to craft Cosmium Catalyst at random. You may cast that card without paying its mana cost.

{T}: Add {R}. Spend this mana only to cast an instant or sorcery spell.
Craft with four or more red instant and/or sorcery cards {3}{R}{R} ({3}{R}{R}, Exile this artifact, Exile the four or more from your graveyard: Return this card transformed under its owner's control. Craft only as a sorcery.)

The Lost Caverns of Ixalan Commander

Craft abilities are activated abilities with costs that have a mana component as well as an additional "materials" component.

Craft abilities are written "Craft with [materials] [mana]", which means "[Mana], Exile this permanent, Exile [materials] from among permanents you control and/or cards in your graveyard: Return this card to the battlefield transformed under its owner's control. Activate only as a sorcery."

If a card that isn't a transforming double-faced card becomes a copy of a card with craft, it'll stay in exile if you activate the craft ability. It won't return to the battlefield.

If the card has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

If the materials required include multiple objects, you may exile some of them from among permanents you control and the rest from among cards in your graveyard. You don't have to choose all permanents or all cards from your graveyard.

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

The back faces of some cards with craft refer to cards "used to craft" it. This refers to the cards exiled as part of the cost of the craft ability of the front face. Those cards are considered to be "used to craft" that permanent as long as they remain exiled and the permanent remains on the battlefield, even if the permanent's controller changes or some of its characteristics change (because of a copy effect, for example.)

You choose whether or not to cast the exiled card as Cosmium Catalyst's activated ability resolves. If you do, you do so as part of the resolution of that ability. You can't wait to cast it later in the turn. Timing restrictions based on the card's type are ignored.

You may exile tokens you control as part of the materials required. However, because they aren't cards and won't stay in exile, any abilities that refer to what you "used to craft" the back faces won't refer to anything.