This has been bothering me for weeks and I still don't have answers.

From what I've been told, disposition affects two things. A wolf's chance to pursue (in addition to agility) and the chance a wolf will hit whatever specific enemy you choose.

Recently, however, the words (wolf name) feels independent, which appears when my wolf's disposition isn't good enough to work to pursue or hit a target enemy, appears when I have no such settings.

What else does disposition affect?

Mainly those two things. Also, disposition ratings (Obedient, etc) are a range. So they may be better or worse even though they have the same rating. If you want the exact number, there is that option.