Will I not get an Extreme Weapon doing them for the first time?

[Nero]

Okay good, so whenever I decide to try my hand at Ifrit EM, even if it's next week, I'll still be able to get one weapon from finishing all three even if there were gaps in-between each fight?

Like the other guys said; yes. And you also do not have to kill them in that order, you could also start with Ifrit, then Titan, then Garuda. But yes, it does not matter how long it takes or when you did accept the quest. Just one thing; Lets say you grab the quest this week and finish it next week, the quest will be locked next week. So you have to sit out until next reset.