What triggers the bad blood between Crawford and Wallace?
Crawford is a know-it-all intellectual, used to humiliating rich kids who haven't had to work for anything. So when suddenly faced with someone at least as intelligent, or perhaps more so, he feels threatened and must do all he can to prove that Wallace is beneath him.
You bring up the possibility of ethnic rivalry. While I think there was possibly some of this implied as well, there isn't any concrete indication of it in the movie. I think the primary factor was Crawford asserting his superiority over others intellectually since he seemed to treat all the students this way.
The following scene is a great example of this conflict.
How did Crawford know about the alleged plagiarism if the story was unpublished?
Answer: it was published in The New Yorker
Crawford: We were hoping you might clarify some points concerning your submission. "A Season of Faith's Perfection." Your piece, correct?
Jamal: Yeah, that's it.
Crawford: It is standard policy to ask students if they wish to credit any source material or acknowledge any other writers when turning in an assignment. Do you wish to do that?
An essay titled Baseball's Best Year...with a subtitle that reads: "A Season of Faith's Perfection." Published in the New Yorker and written by William Forrester.
Your version is quite original, but there is the title and first paragraph to consider. Isn't there? Either you happen to have William Forrester's permission or have you some other explanation?
Jamal: No. That's my paper.
So Crawford recognized the title from when William Forrester published the essay in The New Yorker. Jamal was unaware of this when he turned in the paper, but he had also promised to keep his relationship with William a secret.
Jamal: Why have me rewrite something you published?
William: Be careful.
Jamal: Why didn't you tell me some magazine ran it?
William: Why the hell should it matter? ... You promised me anything we wrote in here would stay in here.
William of course shows up to redeem Jamal by telling everyone in public that he in fact gave him permission to use his work.