Hello all,  I've just identified Tune #2 on Wurlitzer 165 band organ
roll 6529-A, posted at http://www.youtube.com/watch?v=pTZCcX0jFn8 

It's called "Pull The Cork Out Of Erin (and let the River Shannon flow)"
and it was composed by Fred Fisher, with lyrics by Addison Burkhardt,
and published by McCarthy and Fisher of New York in 1917.

According to the sheet music, it was featured by vaudeville superstar
Nora Bayes, and so must have had at least some degree of popularity in
1917.  Here's the sheet music:

  http://purl.dlib.indiana.edu/iudl/lilly/devincent/LL-SDV-150001-01 
  http://mainemusicbox.library.umaine.edu/musicbox/pages/full_record.asp?id=VP_012592 

and here's a recording by Ms. Bayes herself:

  http://www.loc.gov/jukebox/recordings/detail/id/5569/ 

You can hear all of the Irish themes in the music of this song, which
is what tipped me off that it was an "Irish"-themed song from 1916 or
1917.

Enjoy!

Andrew E. Barrett, January 11, 2014