Key to Success – You’re already screwed on this one. In order to get this, you have to specifically not get 100% completion by completing certain Tobli missions and skipping the quest to stop the Ronso.
Invincible – This isn’t like the last one, but if you’re in Chapter 5, it’s probably already too late. You had to talk to a guy with a sphere camera focused on the new Youth League base being built in Kilika in Chapters 1 and 3 and then come back in 5 and view it again and then you can go visit it.
AP Egg – Play some blitzball, or find it in the chocobo dungeon. There’s a map at GameFAQs.
Cat Nip – Since you’re talking about the dungeon in Bevelle – just go do it. The first 60 floors or so aren’t that hard. Just make sure to evade the Elder Drakes, or equip Berserkers with Evade and Counter. You’ll get a treasure chest with this once you beat the 40th floor boss.