Typhlosion

Say a player announced "double Afterburner" for his two Typhlosions in play. However his Discard Pile only has one Fire Energy card for him to use. Does that mean his second "Afterburner" Poke-POWER is used up, or can he use it later in the turn since nothing happened for the second one?

Since you cannot use an Ability for no effect, if the player only has one Fire Energy card in the Discard Pile the first Afterburner Poke-POWER may be used but the second cannot. So since it is not used up the player could use the second one later during the turn if a Fire Energy card somehow ends up in the Discard Pile.

Can Typhlosion's "Afterburner" Poke-POWER be used if there is no Energy in the discard pile, just to place a damage counter on one of my Pokemon?

No. Since you can't attach an Energy from your discard pile to one of your Pokemon, you can't use the rest of the power.

If the Defending Pokemon doesn't have any Energy cards attached to it, does Typhlosion's "Flare Destroy" attack still require Typhlosion to discard an Energy from itself?

Yes, if the attacking Pokemon has any Energy cards attached, it must discard one of them.

When I use Typhlosion's "Flare Destroy" attack, which player chooses the Energy card to discard from the Defending Pokemon?

The player that used the attack gets to choose which Energy cards get discarded from both the Active and Defending Pokemon.