To help organize things, I'll make a list of all the songs, by page number (from the first edition) and opening line:
1) p. 75-7: "We thought that you might like to know": Reprised on p. 250-1 as "We hope that you've enjoyed your stay"
2) p. 85: "When first the light of dawning": Perhaps not a song, just a short inscription
3) p. 87-9: "We'd like to welcome you sincerely"
4) p. 94-5: "I'm supposed to be a princess": Reprised on p. 106-7 as "I suppose you've heard the story"
5) p. 124-5: "Rollin' rollin' rollin'": Identified as "
Rawhide" by Dmitri Tiomkin and Ned Washington.
6) p. 128-30: "From the minute you're born": The evident "title" of this song is "My Own Sweet Tyrannical Way."
7. p. 138-40: "When I was just a little chap, my father said to me": Sounds very Gilbert & Sullivan.
8) p. 161: "Falling apart again": Identified as "
Falling in Love Again" by Frederick Hollander and Sammy Lerner.
9) Also p. 161: "I'm an automaton": May be "
Just a Gigolo" by Leonello Casucci and Julius Brammer, but doesn't quite fit.
9a) p. 162: "Oh, I / Ain't got no body...": These two lines quoted in isolation are a paraphrase of "
I Ain't Got Nobody" by Spencer Williams (best known as part of a medley in which it is combined with "Just a Gigolo")
10) p. 165-7: "Once the screen was really silver": The evident "title" of this song is "Monochrome."
Note: the last line of song 3 is "Stompin' at the Savoy Way." "Stompin' at the Savoy" was a Benny Goodman song, however, its verse structure doesn't match song 3.
The one I'm most curious about is "My Own Sweet Tyrannical Way." Back before I knew there were real songs these were based on, I composed my own melodies for them in my head. Most of them were forgettable, but I came up with a good one for this song, which was actually based on "Hit the Road Jack," but modified to fit the lyrics. It's the only one of the tunes I came up with that I still remember. The real song it's based on probably sounds entirely different...