Okay, I gotta get this settled once and for all...

[Animarelic]

Gear seems to be the biggest issue, make sure your gear is close to or over the quest req. If you managed to get a few tomes or seals buy something. If you are in good standing with your FC buy the gear and pop sanction. Carry some pots as well.

[Anarnee]

The most trouble I had was on my main as DRG. It was HARD. But I appreciated that because it taught me a lot about my job. On most of the rest I had gathered gear for my job that was like 80+ so I had no trouble XD Monk was a breeze, and Warrior too~ Had a little trouble with WHM but I hate healing so...

[Saseal]

How in the world do you NOT die at the final Job Quest mission period? Almost every time without fail, I will usually die due to the fact I'm just not strong enough to handle getting pummeled to death until I get an Echo buff or two.

Monk Quest? Usually end up dead because I get jumped by the addons and too weak to handle the boss.

I can't say much for the others, but for the final Monk job quest, I died a few times also - till I ignored the boss when the adds spawned and downed them first with rockbreaker, etc... I think I may have also kited (after the adds) till second wind/potion came back off CD, too, at one point o_O
It's been a while since then, so I don't remember details, or if the ADDs had healed me or not (I guess it makes sense if they did) - I just remember having a ton easier time after ditching mister monk man for the adds (I don't remember what I was doing before then o_O ....too long ago, too terrible of a memory).

Try plodding Steel Peak to stun the guy to get an additional advantage of being free when going after the adds. I don't think I did this back then (so I'm unsure if it will work), but it's something I would probably attempt if I did this quest now.

[Ashkendor]

I only recall having trouble with the PLD and WHM ones. After that point, I've been using those two classes to farm gear for alt classes. My WAR went into the final quest at i112 so I just kinda stood there and facetanked everything that Gorge threw at me. I remember kiting the DRG around a lot, now that I think about it. That quest is, as previous posters have said, all about knowing when to strike. When he uses a skill that locks him in place, take the opportunity to flank him for the damage bonus. You can stun him too, but that loses effectiveness after 3 stuns.

Definitely make sure you have appropriate gear. When/if adds pop up, make sure to focus them down.