The list is showing some items for my warrior that I cannot actually get from these quests.

Specifically it's saying I need:
Imperial Qiraji Armaments:

• Blessed Qiraji Musket
• Blessed Qiraji Pugio
  Imperial Qiraji Regalia:
• Blessed Qiraji Acolyte Staff
• Blessed Qiraji Augur Staff

All of the quest rewards for these 2 quests are class-restricted. The option to select them doesn't even show up in the UI when you complete the quest.

It's a little frustrating because this causes the 2 Imperial Qiraji items to show up under all of the bosses that drop them, and if those are the only items listed, those bosses wouldn't show up at all.

If you go to turn in the quest at the end of AQ40 it will show zero rewards, but once turned in you still receive all items and transmogs associated with that quest. Blizzard changed it to do this once they added in the option so rewards showed up specifically for your class but still granting appearances.

AH, that was non-obvious. Thanks for the tip!

Yeah, I was confused too when I did it on my warrior back when ATT came out and found that to be the case.