"Eggs for One Hundred"
For this quest, you'll need to backtrack into Naraku and head to the 1st Observation Platform - that big ring where you fought Medusa, remember? Head to the southeastern corner of the outer ring and look out in the distance when prompted; you'll see a bird's nest. Go to it and try to find the egg. You'll be attacked, of course.
|20||Flight||Fire||Force, Light, Dark||Gun (Repel)|
On your side of the battle, you won't find this too hard to contend with. Fire skills should be aplenty by now - or at least Agilao/Maragi Stones - so it won't be too difficult. You could also buy the Flaemis Rounds back in Ueno if you're having trouble. It's not too hard to win without the enemy getting a turn, either - I did it at Level 15.
After getting the egg, return to deliver it back at the Association.
"I Want to Make the City Bright"
Head to the Central Way, which can be found west of the intersection in the Central Underground District. ... And, well, you don't really need a separate section for this, okay? Keep on reading...
"Slay the Pirate Demons"
I only found this in New Game+. Anyways, you need to go beat Pirate Hordes in Ameyoko Way, which can be reached through the vent to the east of the Terminal. The Hordes pop up at random, but are easy to beat if you have multi-target moves.