Music background:

1. The Big Cauliflower
I've been unable to trace this song/tune. According to an advertisement, one songster purported to contain the full music for Clifton's "Robinson Crusoe", but listed the first section as "A Wet Sheet and a Flowing Sea" rather than "The Big Cauliflower". However, the music in RC is definitely not the popular Walton tune for "A Wet Sheet", though it might possibly fit that text.

2. The Lancashire Lass
This song was published at least by 1869. The music is credited to Jesse Williams in a notice for De Witt's Half Dime Music. The lyrics were probably written by George Leybourne, who popularized the song and wrote many others. Sheet music may be found in the Levy collection, and several broadsides exist, including one in the Library of Congress American Memory collection. I'll post the full song later in this thread.

3. I wish I was a Fish
This song was written by George W. Hunt by 1869. It is also known as "Sweet Polly Primrose". The Bodleian has a copy of the sheet music, and the lyrics have been printed in broadsides (see the Bodleian Library online ballads collection) as well as in The American College Songster (pp. 178-9) compiled by S.C. Andrews. I'll post the lyrics later in this thread.

4, 9, 13. Robinson Crusoe
Clifton may have written the tune for the sections titled "Robinson Crusoe"; I haven't traced another source. On the other hand, several songs about Robinson Crusoe were knocking about the music halls before Clifton's extravaganza, and a songster of the day published one version with words "by various authors", indicating perhaps that one song was especially popular and had inspired a number of imitations which quickly became intertwined.

5. Up a tree
This is another Clifton song; he wrote the words and possibly the music.
See this thread.

6. The Calais Packet
The tune is Welsh "traditional", called "Hunting the Hare" (Hela'r Sgyfarnog, though the spelling varies widely). It seems to be primarily a dance tune; the only lyrics I've turned up have been for English art songs.

A popular song called "The Calais Packet" appeared around 1820; at least one source attributed it to William Makepeace Thackeray. It's mainly a recitation with a few sung bits, but they don't seem to match up with the Welsh tune used in the present work.

7. Isabella
Another Clifton song, "Isabella, the Barber's Daughter" (1866), also called "Isabella and Her Gingham Umbrella"--hence the reference to the "umberella" in the current work. Clifton wrote both the words and music.
See this thread.

8. In fact you know my way
I haven't turned up any information on this work.

10. The Fisherman's Daughter
A popular song more fully titled "The fisherman's daughter, that lives o'er the water". The music was written by Samuel Bagnall, but I haven't found an author credit. The lyrics were printed on several broadsides (one placing the latest date at 1866); another, at the American Memory site, has lyrics for both this song and "The Lancashire Lass", mentioned above. The song, with music, may be found in Naval Songs (p. 125) published by Wm. A Pond & Co., New York, 1883. I'll post the full song later in this thread.

11. Paddle your own canoe
Another Clifton song, from 1865. Words by Harry Clifton, music by Charles Coote, Jr.
See this thread.

12. Ten little niggers
Please don't shoot the messenger. This was a minstrel song adapted by Frank J. Green in 1869 from the song "Ten Little Indians". I know of two popular tunes for this song, neither of which matches the tune used in the current work, nor do I know if it is the same tune used by Green. Steve Gardham has told me he has sheet music for the minstrel song, but I don't know how it compares to the present work.
See also this thread.

14. Act on the Square
A dreadful (but dreadfully popular) moralistic/motto song written by "Anthony" with music by Alfred Lee (1866), which was popularized by Alfred Vance, aka. "The Great Vance", aka. Alfred Glenville, born Alfred Peck Stevens. Vance was a clown turned "lion comique". His most popular song was "Walking in the Zoo", noted for containing the earliest known use of the term "O.K." in its current sense. Alfred Lee also wrote the music for that piece, as well as for the song "Champagne Charlie" (among numerous others).

The Bodleian Library has two copies of the sheet music for "Act on the Square": Harding Mus. R 3711 & -2. Lyrics may also be found in broadsides at the Bodleian Ballads site.

15. When I put down dis hoof
I haven't turned up any information on this work. I would guess it is an American minstrel dance tune.