The target nonland permanent card still counts toward the number of permanent cards in your graveyard for the purposes of Squirming Emergence.

Like all spells that require targets, Squirming Emergence will check to make sure its target is still legal as it tries to resolve. Use the number of permanent cards in your graveyard at that time, even though it may be a different number than the number that were there as you cast the spell. If the permanent card's mana value is greater than the number of permanent cards in your graveyard at that time, Squirming Emergence won't resolve.

Cards with the ability word "fathomless descent" have abilities that care how many permanent cards are in your graveyard.

Some descend triggered abilities include intervening "if" clauses (i.e. "if you have [four or eight] permanent cards in your graveyard" in the middle of the ability). Each of these abilities checks your graveyard at the moment it would trigger to see if it does. If you don't have the required number of permanent cards in your graveyard at that time, the ability doesn't trigger at all. If it does trigger, it will check again as it tries to resolve. If you don't have the required number of permanent cards in your graveyard at that time, the ability won't resolve and none of its effects will happen.