My understanding was that there were 2 abilities in madness--a static ability that gave you the option of removing it from the game rather than putting in the graveyard and a triggered ability that allowed you to play it for its madness cost that would trigger if you did remove it from the game.
You're correct here, so let's take a look at the exact wordings from comp rules:
“If a player would discard this card, that player discards it, but may remove it from the game instead of putting it into his or her graveyard�
So, at the time you pay costs for Firestorm, you can discard the Wurm and choose to apply the effect, removing it from the game.
“When this card is removed from the game this way, until that player passes next, the player may play it any time he or she could play an instant by paying [cost] rather than paying its mana cost. When the player passes next, he or she puts this card into his or her graveyard.�
After Firestorm goes on the stack, the Wurm is RFG'd and you have priority. Now is the time to activate LED and get your mana. Even though it has the 'can only be played as an instant' clause, it doesn't use the stack, so you don't have to pass priority to have it resolve. So, you'll get your



with the Wurm still removed, voila.
I thought you had to use the remove from game ability at the time you discard it and would otherwise put it in the graveyard, right? And LED ability doesn't use the stack? So the main effect of its wording is to make it so you can't use it to play spells from your hand?
Correct, correct, and correct. Wizards decided the 'announce a spell, then pay costs with LED once it's out of your hand' play was not intended and too strong, so they fixed it.
I also added a question about timing/turn structure which wasn't there originally
In order to be able to attack/tap a creature you need to have controlled it since the beginning of the turn. Does this mean the very beginning of the turn or just the beginning phase (untap/upkeep/draw)? i.e. If I put a counter on Aether Vial during my upkeep can I Vial out a creature and attack that turn with it?
Beginning of turn is beginning of turn, no getting around that. It has to be there at the beginning of the untap step.
Also: Someone was saying something interesting about Kiki-Jiki. They were arguing that you could play Kiki, then after EOT triggers during your opponent's turn use its ability. Then during your turn you could untap and use it again. Can you do effects that last "until end of turn" at the end of a turn but after EOT triggers like this?
This happens because of a discrepancy between 'at end of turn' and 'until end of turn'. Effects that last 'until end of turn' truly mean that, they are removed during the cleanup step and there's no way to get them to last into a different turn. 'At end of turn' effects are different, because they're actually triggered abilities that go on the stack at the beginning of the end step. The catch here is that the end step is a time when players can get priority, so if you play an ability with an 'at end of turn' trigger after the current turn's end step has already begun, the trigger won't retroactively activate, so it'll have to wait until the next turn's end step to trigger. It's the same idea as putting something into play that has an 'at the beginning of your upkeep' trigger into play during your upkeep - it obviously won't trigger for that upkeep, you'll have to wait for the next one.