Lapras

Can I use Eevee's "Call for Family" to search through my deck if I already have four Eevee in play or in the discard pile?

You can use the attack, but you cannot search your deck since it is public knowledge that you won't find an Eevee because all four copies of the card are already in known areas.

Am I allowed to use Gossifleur's "Call for Family" attack if my Bench is already full?

You can use the attack, but since it's public knowledge that you can't put any Pokemon onto your Bench the entire effect of the attack fails; you can't even search your deck.

Emolga with a Grass energy attached is poisoned and uses its "Call For Family" attack to get Virizion EX with "Verdant Wind" ability. Does it take poison damage or not?

No, because Virizion EX is in play before the end of the turn and Poison damage is taken between turns.

Say my opponent and I each have 1 prize card left, then I use an attack like Pumpkaboo's "Reckless Charge" that KO's my opponent's Active Pokemon but also KO's Pumpkaboo due to "Reckless Charge's" recoil effect. However, my opponent has only one benched Pokemon with 20HP left, while I only have Iron Valiant ex on my bench. Does our game go to a tiebreaker since we both have a Pokemon on the bench, or will I win the game since Iron Valiant ex would be promoted as Active and then "Tachyon Bits" will be activated resulting in the KO of my opponent's last Pokemon?

All effects have to finish resolving before the winner is determined, so you could activate Tachyon Bits to KO the opponent's last Pokemon. You would win the game as you have achieved two win conditions but your opponent has only achieved one.