Lumineon V

Say Gapejaw Bog stadium and Ting-Lu ex with "Cursed Land" Ability are both in play. If I put Lumineon V onto the Bench, can I use its "Luminous Sign" Ability or will it be blocked because it has damage counters on it?

Since Luminous Sign and Gapejaw Bog are simultaneous effects, it is up to the player to choose the order in which they will be applied.

If I use Lumineon V's "Aqua Return" attack against a Defending Pokemon with a Handheld Fan attached, does the Handheld Fan move an Energy from Lumineon V to one of my Benched Pokemon or does the Energy get shuffled into the deck due to "Aqua Return"?

Aqua Return puts Lumineon V into the deck as part of its attack before the Energy can be moved for Handheld Fan's effect. The order of operations is that effects of the Pokemon's attack happen first, and then any triggered effects on the Defending Pokemon happen next.

If my Lumineon V attacks with "Aqua Return" to take my last prize card, but Lumineon V is my only Pokemon in play, do I win because I took my last prize before shuffling Lumineon V into my deck or do I lose because Lumineon V get shuffled in before taking my last prize?

Everything has to resolve before you check for win conditions, and in this scenario both players have achieved one win condition; the game would then be resolved by either Sudden Death or a Tiebreaker Game. However, if the opponent also has no remaining Benched Pokemon then they would lose due to 2 win conditions vs. only 1!