Once you announce that you're casting a spell, no player may take actions until the spell has been paid for. Notably, opponents can't try to change by how much Drag to the Underworld's cost is reduced.

Mana symbols in the text boxes of permanents you control don't count toward your devotion to any color.

To determine the total cost of a spell, start with the mana cost or alternative cost you're paying, add any cost increases, then apply any cost reductions (such as that of Drag to the Underworld). The mana value of the spell remains unchanged, no matter what the total cost to cast it was.

If you put an Aura on an opponent's permanent, you still control the Aura, and mana symbols in its mana cost count towards your devotion.

Colorless and generic mana symbols (, , , , , and so on) in mana costs of permanents you control don't count toward your devotion to any color.

Hybrid mana symbols, monocolored hybrid mana symbols, and Phyrexian mana symbols do count toward your devotion to their color(s).

If an activated ability or triggered ability has an effect that depends on your devotion to a color, you count the number of mana symbols of that color among the mana costs of permanents you control as the ability resolves. The permanent with that ability will be counted if it's still on the battlefield at that time.

The cost reduction ability reduces only the generic mana in Drag to the Underworld's cost. The colored mana must still be paid.