ok first of all, you are looking at this kind of backwards...the idea of the poem is a direct critisizm of both the renaissance era and the seperation of church and state... as far as the duke goes... the duke thinks of his past wife as an object. he did not get his feelings hurt by being ignored by his wife, he was hurt in his pride. He has been used to getting whatever he wants to because he is a duke, a status and symbol of power during that time. He is talking with an ambassador who was sent from the Count of Tyrol because the duke wants to marry his daughter. He walks with the ambassador through his hall of art, but is not apprietiating the art. He expects the Ambassador as well as the reader to apprieciate his status because he is able to afford art. But browning writes this as a critisizm on the duke because the duke doesnt apprietiate art OR the consecration of major. Th duke, in fact, is being revealing about his own psycho-analysis because he treats women like objects. He shares with the Ambassador that he had his last wife killed because she was not the lady she should have been for him. That is because she hired a monk (who swares himself to god and is not open to sex or relations) who is also a painter, to paint the duchess, his wife. Then he assumes because she is smiling in her painting that they are flirting. He calls his wife permiscuis and says that she is betraying his name and status. The Duke is telling this as a warning for the Ambassador to tell the Count's daughter. To tell the new girl that she better keep in line and be what he wants. That he deserves nothing less and that he will bow down to no one. The reference to stooping is that he will not stoop for any woman. That she did not just obsess and brag over him because she loved "the sunrise" and eating cherries. He had outlandish views about love and claimed she did not respect what she was. Mean while, back in those times, he may have been in his mid thirties wheras the wife is 14 or 15.  she was just excited and kid like about how people were treating her special. But the Duke is so concieted that he thinks her smiling is not apprietiation but shame to him. He is arrogant and self absorbed. The painting is hidden behind a cutrain which no one could see but him. He is ashamed of her, which is why he killed her. He is empty in his heart and just looking for the next girl to wear on his sleeve as a symbol os status...and that is what i have learned and analysed. 

There is also the possibility, since the Duke is talking to the man who is there to negotiate a new wedding contract, that "Neptune taming a sea-horse" is meant to imply that the Duke's new wife will receive no better treatment than his last wife should she be of a similar or independent nature.  The Duke obviously views women as either wild and ungovernable beings that must be subdued, or as naive, unintelligent creatures that must be trained...a thoroughly detestable character!

In my opinion, I dont think he beat her, I think that his given commands were to kill her though. She was always so easily pleased with such simple matters that this made him jealous that her attention was all different directions and not focused on him. As well, it was the fact that she was talking and regarding " lower people " of wich the duke would never talk to. She finds enjoyment in simple things in life and was a kind-hearted person. The Duke did not like her characteristics so he wanted a new wife of which could suit them.

He loved her, but she couldn't stay faithful. Her eyes were always straying. He had her killed, and kept that picture behind a curtain so he could control who could see her. Since he couldn't when she was alive. He was a very proud man of his name and of his art couldn't handle a wife that had eyes for other men.