Cresting Mosasaurus's last ability triggers if you cast it from any zone. It doesn't trigger if you put Cresting Mosasaurus onto the battlefield without casting it.

The mana value of a creature spell with emerge isn’t affected by whether its emerge cost is paid. For example, if you cast Decimator of the Provinces for its emerge cost and sacrifice a creature whose mana value is 3, Decimator of the Provinces’s mana value remains 10.

Emerge represents two static abilities that function while the spell with emerge is on the stack. “Emerge [cost]” means “You may cast this spell by paying [cost] and sacrificing a creature rather than paying its mana cost” and “If you chose to pay this spell’s emerge cost, its total cost is reduced by an amount of generic mana equal to the sacrificed creature’s mana value.”

Once you begin to cast a spell with emerge, no player may take actions until you’re done. Notably, opponents can’t try to remove the creature you wish to sacrifice.

You may sacrifice a creature with mana value greater than or equal to the emerge cost. If you do, you’ll pay only the colored mana component of the emerge cost.

The mana value of the back face of a double-faced card is the mana value of its front face. The mana value of a melded permanent is the sum of the mana values of its front faces. A creature that’s a copy of either has a mana value of 0.

A creature’s mana value is determined solely by the mana symbols printed in its upper right corner (unless that creature is the back face of a double-faced card, is a melded permanent, or is copying something else; see below). If the mana cost includes , X is 0. If it’s a single-faced card with no mana symbols in its upper right corner (because it’s an animated land, for example), its mana value is 0. Ignore any alternative costs or additional costs (such as kicker) that were paid as the creature was cast.

If you sacrifice a creature with in its mana cost, that X is 0.

The creature chosen to be sacrificed is still on the battlefield as the cost of the emerge spell is determined and as you activate mana abilities to cast the emerge spell. Its abilities may affect the spell’s cost, be activated to generate mana, and so on. However, if it has an ability that triggers when a spell is cast, it will have been sacrificed before that ability can trigger.

Colored mana components of emerge costs can’t be reduced with emerge.

You may sacrifice a creature with a mana value of 0, such as a token creature that’s not a copy of another permanent, to cast a spell for its emerge cost. You’ll just pay the full emerge cost with no reduction.

If you cast a spell for another cost “rather than paying its mana cost,” such as an emerge cost, you can’t choose to cast it for any other alternative costs. You can, however, pay additional costs, such as kicker costs. If the spell has any mandatory additional costs, those must be paid to cast it.