1.) The Duke is NEVER, for even a moment of the duration of this poem, "witless", though "shrewd" does not not exactly cover it, not completely, anyway. Sure, he is basically stage-managing the entire incidnent of displaying the late wife's portrait, with the intention of intimidating the emissary who has almost concluded the negotiations for his new marriage.
However, in the course of relating the story of the first marriage, the Duke reveals the embarrassing and, ultimately, horrible story of its collapse, which may not have been his true intent. He thinks he is merely making typical husbandly complaints, which, because he is a man with great power, he is able to resolve in a far more decisive manner than the common husbands of his time.
Instead of being witless or shrewd, he is reckless--- he as much as confesses to having his first wife MURDERED, to the man who is responsible for bringing him his SECOND wife. The Duke indicates that a similar fate is in store for the next hapless woman who refuses to be "objectified"--- he even points out what art form will likely memorialize a subsequent obstinate bride--- a new commission for that scultptor in Bronze, Claus of Innsbruck. He makes subtle threats to the emissary, insisting on accompanying him and watching him closely, as they are about to return to whatever social gathering for the betrothal is going on downstairs.
Yet, doesn't he think the emissary won't eventually report on the frightening trap the Count's daughter is being sent into, even by a secret message if necessary? Daughters of nobles and royalty in those days WERE usually regarded as mere pawns to advance their fathers' alliances and profits. But even royalty is human, and most loved or at least liked their daughters; certainly most would not have wished to put their girls into situations where they'd wind up murdered for the "transgression" of smiling and behaving good-naturedly with everyone, being fond of a pet, enjoying compliments and an impulsive gift of fresh cherries!
Still, one must remember, family members of the latter wives of Henry VIII of England, at around the same period, SCHEMED to put their daughters and sisters into the English King's sights, in spite of the terrible things that happened to most of them, and despite the growing deterioration of the King's condition. One can only hope that the Count isn't THAT type of father, and that the emissary would "do the right thing" by warning the gentleman to save both his daughter AND the money due for the dowry.