Paddy at the Theater was presumably set to one form or other of the Derry Down tune. There are several in the DT; the one with KING JOHN AND THE ABBOT OF CANTERBURY would fit the bill:
Remember the Glories of Brian the Brave was written by Thomas Moore, and set to the tune Molly MacAlpine, of which variants may be found under a great many names; for some of them, see: Tune Req: Carolan's Dream.
The Memory of the Dead was written by John Kells Ingram (b. 1822) with music by W.E. Hudson. I'll put up a midi once you've posted the text.
Mary of Tipperary (The Irish Milkmaid's Song) was written by Samuel Lover. Sheet music (1857) is available at The Lester S. Levy Collection of Sheet Music:
MARY OF TIPPERARY. THE IRISH MILKMAID'S SONG.
I feel sure that I've seen O'Reilly the Fisherman recently, probably here; but I'm blessed if I can find it.