Why did Denham drop his boxing gloves at Braithwaite's feet? What did he mean to indicate: a challenge, contempt for his teacher, or reluctance to fight?

Denham drops the gloves at Braithwaite's feet as a symbol of contempt. He had already issued a challenge—which Braithwaite refused—and didn't respect Braithwaite as a teacher. Braithwaite ultimately chooses to accept the fight because he knows that if he doesn't, his other students will think less of him.
When Braithwaite takes the fight, he wins. He punches Denham and injures him badly enough that the fight is over. He knows he only got lucky and Denham was the stronger boxer. Even though he acknowledges this to Denham, it still changes the boy's attitude toward Braithwaite and makes teaching the class easier. This leads to Braithwaite appreciating his job and each of his students more than before.
Even though Denham maneuvered Braithwaite into boxing him, ultimately the act makes the classroom situation better for everyone involved.

Denham is looked up to by the other boys as the unofficial leader of the class. Inevitably, this puts him at odds with Braithwaite, and a power struggle ensues between teacher and pupil, which culminates in the famous boxing match episode. As Braithwaite becomes more popular and respected, Denham starts to see him as a threat to his power over the other pupils in class. Moreover, he resents Braithwaite for ripping up his pin-up and feels the need to reassert his authority in class.
A boxing bout seems just the thing for Denham to settle scores. At first, Braithwaite refuses. It's then that Denham drops his boxing gloves at Braithwaite's feet. The gesture is loaded with significance. As well as being a challenge to Braithwaite's authority as a teacher, it also shows Denham's contempt for what he perceives to be Braithwaite's cowardice. Braithwaite understands this, which is why he refers to the looks of disappointment and disgust on the faces of the other boys when it seems like he's chickened out.