Dragapult VMAX

If my opponent uses N's Zekrom's "Shred" attack and does enough damage to KO my Mega Hawlucha ex, do I get to flip a coin for its "Tenacious Body" Ability?

The full damage will be done by Shred, but then if the coin flip for Tenacious Body is heads 10 HP is restored to Mega Hawlucha ex AFTER all the damage is done. Mega Hawlucha ex would not be KO'd.

Does Pyroar's "Intimidating Fang" Ability reduce damage from "Shred" attacks?

Yes, it does. The effect of Intimidating Fang is on the Attacking Pokemon (not on Pyroar), so Shred's condition of "any effects on your opponent's Active Pokemon" does not apply.

If I use Fantina and then my Flying Pikachu V with Big Parasol attached uses its "Fly" attack and successfully lands heads, would my opponent's Giratina V's "Shred" attack do any damage to Flying Pikachu V or not?

Shred isn't affected by "any effects on the Defending Pokemon"; so the effects from Fantina and Big Parasol and Fly would all be ignored, and Shred would do damage.

If Vileplume's "Allergy Panic" Ability is in effect, does Rayquaza's "Shred" attack consider a Defending Pokemon's weakness as x2 or x4?

Shred isn't affected by "any effects on the Defending Pokemon", but the term "any effects" does not include effects that alter Weakness or Resistance. So in this case the Defending Pokemon's Weakness would be x4 due to Allergy Panic.

Does Giratina EX's "Shred" attack ignore effects such as Weakness and Resistance?

No, Weakness and Resistance are not considered effects; they are basic game mechanics and are not ignored by Shred.

Would Giratina EX's "Shred" attack allow my opponent to knock out my Shednija (with "Empty Shell") and take a prize since the attack says its not affected by any effects on the Defending Pokemon?

The damage done by Shred is not affected by Empty Shell, but that has nothing to do with whether you take a prize or not. You still get no prize because of "Empty Shell".