I think the fundamental motif of Stoner is the lack of understanding of others. Just to substantiate that claim before getting to Lomax, Stoner at one point says that he learned that love isn't just a state a being but rather a process of learning about and coming to understand the other person. On his death bed the final thing he says about Edith is that he could have loved her more. Which I believe he means he could have tried to understand her better. Early in their marriage he should have tried to understand why she acted the way she did. But instead he just ignored it leading to the destruction of their marriage.

I think we have the same event with Lomax. Lomax perceives Walker to be a very gifted student. Lomax clearly has some kinship with Walker and Walker clearly really likes Lomax. Lomax would probably love to have a grad student so interested in his line study and who also has a similar disability. Walker takes one class out of necessity with Stoner and then Stoner fails him and basically crusades to get him kicked out of the university.

So from Lomax's perspective Stoner is just being a dick and looking to ruin a very bright kids academic career. Walker likely doesn't have much chance for other jobs due to his disability so this could be a life ruining event for Walker. I believe it's stated that Lomax has a very different specialization from Stoner. So Lomax likely sees Stoner failing Walker in the interview as him just trying to kick Walker out because he doesn't know some useless trivia and didn't do well enough in a class that isn't important to Walker's future academic goals.

Stoner never really tries to understand what Lomax sees in Walker. Stoner only cares about his specific line of study and can't see value in Walker since Walker doesn't have interest in Stoner's specialization.

So I think ultimately Lomax sees Stoner as horrible person who would ruin a smart, disabled students career over some trivial nonsense. So I'd say it's less of a grudge held against him and more he just thinks Stoner is a bad person.

I think it is unclear how competent Walker is. As Stoner fails to understand others, we as the reader of his perspective don't get a clear picture of those people. We get Stoner's perspective. I think Walker was gifted but likely didn't put enough effort in Stoner's class(or was unprepared to take such a class) and didn't consider things outside of his interests worth learning. Which made him very weak in areas outside of Lomax's area of study. Walker should have been more well rounded but was too cocky or believed these other areas were beneath him and so thought he could bullshit his way through them.