So, I recently picked up the pauper version of Burn and I have a question about using needle drop. If I have a Firebrand Archer in play, can I use Needle Drop prior to other damage being dealt? I'm assuming no, because Needle Drop doesn't have a legal target as you cast it, but on the other hand the Firebrand Archer's ability would mean it has a legal target as it resolves... Just want to be sure I'm not missing a potentially useful interaction for the deck.
Your initial assumption is correct. You can't legally cast a spell that targets without choosing the required legal target(s) as you cast it. If your opponent hasn't already been dealt damage, you can't cast Needle Drop targeting them. So you can't do what would trigger Firebrand Archer.
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The target you choose for Needle Drop must have already been dealt damage this turn by the time you start casting Needle Drop (C.R. 601.2c, 114.1a, 114.4). However, if an effect lets you change Needle Drop's target (which neither Needle Drop nor Firebrand Archer lets you do), that target could be changed to another creature, player, or planeswalker that was dealt damage this turn, even if the damage happened after Needle Drop is cast and before it resolves (C.R. 114.7, 114.4).
EDIT (after comment 4 was posted): Correctness edit in view of changed text for Needle Drop.
EDIT (Apr. 28): Edited further since Dominaria was released.
EDIT (May 18, 2019): Edited.
EDIT (after comment 4 was posted): Correctness edit in view of changed text for Needle Drop.
EDIT (Apr. 28): Edited further since Dominaria was released.
EDIT (May 18, 2019): Edited.