So far, can't find anything much on Hays' "Away Down in Dixie's Land" beyond what is in the Public Domain website. The song apparently never caught the public's attention.

Dan Emmett had the first performance of his song in 1859 or before. The W. L. Hobbs 1860 arrangement was for the piano, not minstrel use, and deletes some of the early "Negro" lines which can be found in the book and Ms of Galbreath, C. B., "Daniel Decatur Emmett, Author of 'Dixie'," published by Heer in 1904, and manuscript in Ohio State Archaeological and Historical Library. This has been reported before in thread 25204: Emmett . Just how long before the 1859 performance this was written is not noted on the University of Toronto website. The 1893 letter written by a man who knew Emmett and one of the Bryants', linked above, suggests that the song was first performed in 1859. Hobbs apparently had nothing to do with the first version(s) of Emmett's song and its minstrel performance.
Also not seen is the book by Nathan, mentioned above by Masato, which may have some newer researched evidence. These books and Ms should be checked before drawing further conclusions here about Emmett's work. Newspaper reports may also be helpful.

Unless someone is willing to go to these books and old newspapers, and also look for possible archives of Hays papers, little more can be investigated here. Internet possibilities have been exhausted. Masato may have access to the books. I don't believe any of us has the time or travel money to do original research.

Murray MacLeod and others, led by the title of this thread, have asked about the origin of the word Dixie. For them, some evidence and speculation has been posted.
One addition, perhaps posted before- "dixie" is a British Army name for a cooking pot or "billy," picked up from the days of the British Empire in India. Now if someone is willing to make up a story about how a cooking pot provided the name for the South...!