Sky Pillar says, "Prevent all effects of the opponent's attacks, including damage, done to Benched Pokemon (both yours and your opponent's)". So does this mean if I use an attack like Zapdos' "Raging Thunder" that I do not apply the bench damage to my own Pokemon?
No, only bench damage done from an opponent's attacks is prevented by Sky Pillar. If your attack does damage to your own Benched Pokemon it is not prevented.
If Voltorb is attacked by Blastoise EX's "Rapid Spin" which pushes it to the Bench after doing damage, do you still flip for Voltorb's "Destiny Burst" Ability or not?
Yes. Destiny Burst checks for two things: Voltorb was damaged by an attack while it was your Active Pokemon, and Voltorb was Knocked Out by that attack. So even though you don't check for a KO until the attack is completed, the criteria for using Destiny Burst has been met, so it can be used. This holds true for any similar Ability that triggers when the Pokemon is Active and gets damaged by an attack.
Source:
XY FAQ (2014-02-06), TPCi Rules Team (2014-02-06), TPCi Rules Team (2018-08-09)
Say I have taken 5 prize cards during the game, but then my opponent uses Naganadel GX's "Stinger GX" attack to bring both players' prize count back to three prizes remaining. So if I then use Stakataka GX's "Assembly GX" attack on my next turn, how many prizes would I count when calculating the amount of damage to be done?
Even though the Stinger GX attack added cards back into your prizes, you have still taken five prizes during the game. So Assembly GX would do 50 damage plus 250 (five times 50) for the prize cards you have already taken.
Does Venusaur's "Jungle Totem" Ability boost the damage output of Rayquaza GX's "Dragon Break" attack since "each Basic {G} Energy attached to your Pokemon provides {G}{G} Energy"?
Yes, just because the Basic Energy card is now providing two units of energy doesn't change the fact that it's still Basic Energy.
If I use Seedot's "Harden" attack which prevents any attack damage that is 40 or less during the opponent's next turn, does that come before or after calculating for Weakness/Resistance?
Weakness & Resistance are factored in before determining whether Harden blocks it or not. So if a fire Pokemon's attack does 30 damage to Seedot and then +30 more is added for Seedot's Weakness, Harden would not prevent the damage. However, if the fire Pokemon's attack does 20 damage to Seedot and then +20 more is added for Seedot's Weakness, Harden would prevent the damage.
Source:
Celestial Storm FAQ (2018-07-20), TPCi Rules Team (2018-08-02)