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.

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.

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.

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.)

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.

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.

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.