There is no suggestion in the poem that the painting is positioned on the grand staircase. The Duke asks his visitor "to sit and look at her," which would not have been possible on any staircase. Furthermore, when he invites the visitor to go downstairs to "meet the company," he calls attention to a statue of Neptune. Most likely they are in some fairly large room filled with paintings and statues, probably the Duke's library or private study. You are astute in calling attention to the fact that the painting is kept covered by a curtain. It is a matter of conjecture why the man would have his wife killed but keep her portrait and then cover it with a curtain, but it seems likely, judging from his character, that he considered the painting too valuable to discard or destroy, while at the same time he didn't care to be continually reminded of his dead wife and what he had done to her. In fact, the painting may arouse the same ill feelings that he had toward his living wife when he ordered her death. It's odd that he can look at the painting with his visitor and not express any sense of guilt or regret. The painting only seems to remind him of what he didn't like about her--the fact that she was too happy and too friendly.
The duke has covered the painting with a curtain to try to forget his last duchess' existence, probably because he had her annihilated. The curtain symbolizes a sign of remorse, which technically means that the painting was covered with sadness and regret. However, the king does not want people to know about his other side: his warmth, but as a reader, you can tell that he has a soft side because not only does he regret his decision about having his wife killed, but he killed her in the first place because he was jealous that all the attention wasn't on him.
Sorry, its a little confusing, but I hoped this helped.