Now I know you just need to defeat the monster to get this fragment, but in the bestiary page some monsters have a check mark and some don't.  I also noticed that some have an X to the right of their name while others just have  a little smily face.  Do you need a check on each of these monsters, or merely need the name to show up on the list for the fragment?  And does anyone know what the check marks mean?

I think the smiley face means it's tameable and X means it's not, and the check mark means you've got 100% of the information for that creature.

And no, you don't need the check marks for the fragment.

You need to basically fight every monster in the game, and that includes the 3 different versions of Adam in the tower annoyingly! Killing the Long Gui is going to be a task and a half... 4 MILLION health o_O

Thank you.  I searched various sites for this information but I think you are right.  

I actually beat the Long Gui without much trouble and I accidentally got the Giant's Fist trophy while doing it.  I used basically the same strategy I used in XIII, it just took about 3 times as long.  The 3 forms of Adam though are going to be annoying especially since this is the last fragment for Fragmented trophy.