Does Browning imply any connection between the Duke's art collection and his attitude toward his wife?
Browning definitely implies a connection between the duke's art collection and his attitude toward his wife. This poem is all about possession and ownership. The duke proudly points out the painting of his last wife on the wall of his home to the visitor in the opening lines of the poem. With displeasure and suspicion, the duke notes that he was not the only one to make her blush in the same way that she does in the painting. He is an extremely jealous man and views his wife as a possession, much in the same way he would view his art collection.
Note that the duke has a curtain placed over the duchess' painting so he can cover and unveil it at his choosing; then he does not have to share her countenance with others if he does not wish to do so. During her life, the duke could not control his duchess at will; she gave the same favoring smile to him as she did for others, like the white mule or the "officious fool." The duke feels that he ought to be able to control and direct her smiles as he would be able to organize and display his art; he perceives her as merely another addition to his collection, and when she fails to meet his high standards for comportment, he orders her removal, just as he would for a piece of art that displeased him. Now, in her portrait, he can finally control her as he originally wished, free to direct her smiles and display her beauty at his convenience.