Who is the Duke addressing in "My Last Duchess," and what is this person's business in Ferrara?

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The person the Duke is addressing is not named, but he is in the Duke's palace in Ferrera for the specific purpose of discussing the dowry to be paid by the father of the Duke's fiancee, who is a Count but is not named either. In the last lines of the poem, the Duke says:

Will't please you rise? We'll meet
The company below, then. I repeat,
The Count your master's known munificence
Is ample warrant that no just pretence
Of mine for dowry will be disallowed;
Though his fair daughter's self, as I avowed
At starting, is my object.

Evidently the Count himself and some others who are visiting with him are in the rooms below. The Count's representative has been invited upstairs to a private room, ostensibly to view some of the Duke's art collection but actually to discuss money. The Duke use of the words "I repeat" and "At starting" indicate that they came upstairs to talk about money and that money is still the uppermost consideration at the termination of the interview. The Count and the Duke are too aristocratic to discuss the dowry face to face.

When the Duke asks, "Will't please you rise?" it would appear that the Count's envoy is so appalled by what he has heard during the monologue that he can no longer stay in the same room with this brutal man. It may be that the envoy intends to go downstairs and advise his master that he should by no means consider marrying his daughter to this wicked, greedy man. When the Duke says, "We'll meet the company below, then," and a few lines later, "Nay, we'll go together down, sir," it signifies that the Count's representative has abruptly risen from his chair and is leaving the room without even saying goodbye. The Duke apparently has to hurry after him in order to accompany him down the stairs. He tries to delay the fleeing man by saying:

Notice Neptune, though,
Taming a sea-horse, thought a rarity,
Which Claus of Innsbruck cast in bronze for me!

Since the question of the dowry has not been settled, the Duke would appear to be trying to save face by resurrecting the fiction that the two men really only came up here to look at works of art. With the dowry question still unresolved, the Count can hardly go ahead with discussing arrangements for the marriage. The visit might end--we should hope!--with the Count and his party returning to their own home and then with the Count and his wife desperately concocting some excuse for breaking off their daughter's engagement.

This outcome can only be inferred from the abrupt and deliberately rude behavior of the Count's representative, since we know nothing about what he might have said to his master. But it would seem that the Duke has cut his own throat, so to speak, by his self-revelations in his monologue. Evidently he became lost in his own memories and related emotions while looking at the portrait of his presumably murdered wife --

Looking as if she were alive.

To whom is the Duke of Ferrara in "My Last Duchess" speaking?

As the poem "My Last Duchess" opens, the Duke shows a visitor a painted portrait of his former wife. Line 7 groups the hearer in with "strangers," so we can assume the visitor has never been there before. The man, whom the Duke addresses as "Sir," has evidently remarked about the "spot of joy" in the Duchess' cheek, as others have done. But it is not until line 49 that the reader gets information about why the visitor is being given this tour of the Duke's art gallery. 

The Count your master's known munificence
Is ample warrant that no just pretense
Of mine for dowry will be disallowed;
Though his fair daughter's self . . .
Is my object.

From these lines the reader at last discerns that the Duke is speaking to a representative of a Count, arranging a marriage between the Count's daughter and the Duke. Interestingly, and somewhat humorously, it appears that the Duke's rant about his last duchess, and his implied treatment of her, causes the Count's representative to want to hightail it out of there. The Duke has to restrain him from making a hasty exit by saying, "Nay, we'll go together down, sir." That reaction gives the reader hope that the Count's daughter will not become the Duke's next duchess.