For the Spoiler-hungry people:
1. The Bad Guy
The main enemy that Tweight faces off against in this arc is Vallen.
2. Motivations
As seen/hinted at in this chapter, Vallen and Ix used to be friends (or on friendly terms), though this changed when Frau died in the Great Hero-Demon War. While Ix was able to move on from his sister's death, and decided to fulfill her wish of establishing the Common Course, Vallen couldn't. Instead, he cursed the weakness of the Kingdom, which he saw as responsible for Frau's death. In his mind, the weakness of the Kingdom stemmed from their complacency; from living in peace. To address this, he wanted to instill a sense of crisis in the Kingdom, so that the Kingdom would not be complacen; so that more lives like Frau's wouldn't be lost.
3. Mira
Mira is there, on the orders of the President, to prevent others from interrupting Tweight and Vallen's "duel". The purpose of the "duel", as she later tells the trio, is to determine who should participate in War Game.
4. The "Duel"(with context)
Ix already knew that Vallen was doing, yet couldn't bring himself to stop it (even though he and Vallen are equally strong), as it wouldn't get rid of the root cause: Vallen's grudge against the weakness of the Kingdom. So instead, he allows Tweight to duel Vallen, since to Vallen, Tweight is a student of the Common Course, which he vehemently opposes (as it conflicts with his ideals). If Vallen loses against Tweight, he is forced to admit that perpetually living with a sense of crisis does not necessarily make one strong, and that peace is necessary. Whereas if it had been Ix who duelled Vallen, nothing would have changed, as they already oppose each other.