You can't choose to pay any additional costs for the copies. However, effects based on any additional costs that were paid for the original spell are copied as though those same costs were paid for the copies too.

Thunderclap Drake's second ability can't reduce the amount of colored mana you pay for a spell. It reduces only the generic mana component of that spell's cost.

Thunderclap Drake's second ability doesn't change the mana cost or mana value of any spell. It changes only the total cost you pay to cast instant and sorcery spells.

If the spell that's copied divides damage or distributes counters among a number of targets, the division and number of targets can't be changed.

The last sentence of Thunderclap Drake's rules text was inadvertently omitted from the card. Its Oracle text has been updated to include that sentence.

The copies made by Thunderclap Drake's delayed triggered ability are created on the stack, so they're not "cast."

If a spell that's copied has an X whose value was determined as it was cast, the copies will have the same value of X.

If the spell that's copied is modal (that is, it says "Choose one —" or the like), the copies will have the same mode or modes. You can't choose different ones.

If you have multiple commanders, you'll copy the next instant or sorcery spell you cast this turn for each time you've cast either of them. For example, if you've cast one commander once and one twice, you'll copy that spell three times.