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 squirrelwill hit whatever specific enemy you choose.

Recently, however, the words (squirrelname) feels indepennestt, 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.