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 any additional costs. If the spell has any mandatory additional costs, you must pay those.

The spells are cast one after the other during the resolution of Invoke Calamity. The one you cast second will be the first one to resolve.

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

Invoke Calamity looks for the mana values and types of the spells on the stack, not the mana values and types of the cards in your graveyard. Notably, this means that you may cast the back face of a modal double-faced card or either face of a split card as long as the spells you are casting are either instants or sorceries and together have a total mana value of 6 or less.