You do know that event cards are "fought" for like any other card? Consequently, whoever "wins" it gets to use it. They don't just "activate" when drawn.
which should answer your other questions.
I must have misunderstood something in the rules. Are you saying that if I claim the objective card like any other objective card that I am allowed to use the card to activate the special effect in some later turn? For instance, if I claim the "Live Benefit" Objective card during the Debriefing Phase of Turn 1, I can use the "special ability" during the Influence Struggle Phase of any later turn?
That all seems correct. Then if I use the card's "special ability" the card is discarded so I no longer retain the points won for that objective and I remove the points from my score. Right?