Noivern

Player A attacks and flips a heads for "Agility". Then Player B uses Single Strike Urshifu VMAX's "G-Max One Blow" attack; would this ignore Agility?

In this case, Agility would be considered an "effect on the Defending Pokémon" and therefore G-Max One Blow would be able to overcome Agility.

If I flipped heads on Seadra's Agility attack, would that prevent Cradily's Spiral Drain from removing damage counters?

Agility only prevents damage and effects done to Seadra, not other effects that may extend to other Pokémon.

When using Agility in Team Play, can you specify which of the opposing team's Active Pokemon is affected by the Agility effect, or does it HAVE to be the Defending Pokemon that you are attacking with Agility?

It does have a you or your in it, so you would be able to select which opponent's active Pokemon it affects.

In Team Play, when an attack like Kakuna's "Secrete Poison" or "Agility" specifies the phrase "during your opponent's next turn", does that count for both opponents or do you have to choose one of the opponents to have the effect during their turn only?

You would have to specify which opponent.

If a Raichu (or whatever) does Agility and flips heads, does an attack that would do damage to benched pokemon, or some other effect that does NOT do damage or something to Raichu work?

Yes it would, while all attacks are considered against the defending Pokemon (which is why something that prevents attacks would stop it in this case), Agility only prevents the effects of that attack if they affect Raichu only.

If my Pokémon uses Agility and get heads then on my opponent's next turn he benches his Pokémon, does that [Ed. Note: Agility's effect] end? If that Pokémon was Knocked Out, would Agility still be on the next Pokémon that he sends up?

Since the Agility Attack references Rapidash, only Benching or Evolving him ends the effect. Also since it only refers to Rapidash it would still be in effect if the other player benched.

When using Raichu's Agility attack it states that if heads, Raichu will not receive damage until after the opponent's turn. If Raichu is poisoned and then gets heads, will he take damage from poison at the end of his turn?

Yes, Agility is referring to effects of attacks on Raichu during your opponent's next turn.

Does Fearow's Agility attack block Kangaskhan's Fetch?

Actually it does not; Fearow's Agility says stops damage done to Fearow, and the Fetch just draws a card. This is the new ruling from R&D.