The Japanese Doll Dancing--and Singing--

Japanese "plots" for comic and light romantic songs abounded from the mid-1880s to the 1920s. A few of them even had Japanese dolls....

 
In the beginning, of course, or at least in 1885, was Gilbert and Sullivan's The Mikado, with its "Three Little Maids from School" and its "Gentlemen of Japan":

If you think we are worked by strings,
    Like a Japanese marionette,
You don't understand these things:
     It is simply Court etiquette.
          Perhaps you suppose this throng
          Can't keep it up all day long?
          If that's your idea, you're wrong, oh!

The idea that there was somethig  puppet-like or doll-like about the Japanese themselves seems to recur often in novels, plays, and songs about the Japanese
...and so we find that Japanese dolls can speak like Japanese ladies....

 In 1886, American musical theater had seen The Little Tycoon, An American-Japanese Comic Opera by Willard Spenser with songs like "Love Comes Like a Summer Sigh," "Tell Me, Daisy," and "Heel and Toe." It's not clear what the plot was, but it may have involved a young samurai visiting the U.S. (NB the Historic American Sheet Music site lists the date as 1882, but 1896, the actual date on the music, seems more likely; the "Tycoon" was the Shogun who served the "Mikado" or Emperor).

In 1896, Sidney Jones's musical The Geisha, or The Story of a Japanese Tea-House opened in London.  (see below)

In 1911, The Mousmé (Monckton/ Talbot/Percy Greenbank/Wimperis), an operetta, opened in London. Midi files and the libretto can be downloaded from the Edwardian Musical Comedy page. It is a story set entirely in Japan, though one of the heroines is English on her mother's side, and it involves a double romance set against the military ambitions of Japan in the East.
Of the musical plays mentioned above, Sidney Jones's The Geisha was the most popular, and it is still produced and recorded, despite its "politically incorrect" lyrics (see below!). The plot involved the clash of cultures, with British sailors and ladies visiting a Japanese tea-house run by a Chinaman assisted by a young French woman. A Madame Butterfly romance is avoided, thanks to a remarkable number of disguises for the young ladies. The entire libretto is online.
There is also an online Biography of Sidney Jones which focuses on The Geisha.
This is the second verse of The Geisha duet (by a British naval officer and a geisha) "The Dear Little Japppy Jap Jappy."  Though it doesn't mention dolls, the idea of a language barrier is  suggested in some of the doll songs below. In the first verse, a Jolly Jack Tar meets a Japanese girl and falls in love with her; in the third, he goes back on board his ship and she marries a "Japanese chappy chap chappy":

 They walk’d in the shade of the trees
  In the garden of fair Nagasaki,
  And her cheeks they were pink
  At the nautical wink
  And the maritime manners of Jacky!
  Though the Tar couldn’t speak Japanese,
  Yet in English he asked her to marry,
  Then she crept to his side
  And her fan opened wide
  As she murmured: “Hai! Kashikomari!”
  But he knew not a scrappy, scrap, scrappy
  Of the language of Jappy, Jap, Jappy
                      Had she told him to go
                      With a Japanese “No!”
  Or with “Yes” made him happy, hap, happy?

 Later in The Geisha, Molly, Fairfax's fiancée, turns up, quite aware that he has been flirting with a geisha. Presumably "nurse" here means something like "cuddle."

Molly.   Now  take  me into the doll’s house.
Fairfax. It’s not a doll’s house, it’s a Tea House.
Molly.   A  Tea  House  with  live dolls  in  it!  And I believe you’ve been nursing some of them!
Fairfax. Oh no, I haven’t! I’m too old for dolls!
Molly.   You  usen’t to be. Do you remember when you used to nurse mine, and then try to nurse me?
 

  Molly.   When I was but a tiny tot
    My dollies were a lovely lot,
    For one, a lady born and bred,
    Could shut her eyes and move her head.
     “Papa!” “Mama!” another talk’d,
      And when you wound her up – she walk’d,
      But more than any other toy
       I loved a little drummer boy.
The Jap Doll 

Sheet Music, 1898, US/UK
(reprinted in several formats)

by Jessie Gaynor, words by Alice C. D. Riley

I come from the flow'ry kingdom that lies far across the sea, 
And I'm just as nearly homesick as a little doll can be 
For they talk about my funny clothes, and never think of me; 
I'm a very lonesome dolly. 

So I sit up here in splendor on the mantel shelf so high, 
And I watch the Dresden china boy who's guarding sheep close by, 
And if someone doesn't take me soon, 
I'm 'fraid I'll have to cry, 
I'm such a lonesome dolly

I'm a slant-eyed doll from Japan so far away, 
I've a red silk sash and a paper fan so gay, 
But you're not allowed to touch me, 
Unless someone says you may. 
I'm a very lonesome dolly.

 From a list of ragtime compositions on Doc Wilson's Turn of the Century Stuff website:

Japanese Love Song (Clayton Thomas) 1900
Japanese Rag  (Mose Gumble)  1901
Japania--Japanese Two-Step (H.E. Dean) 1906
Japanimo (Arthur D. Porter) 1906
Jappy Johnny--Japanese-American Intermezzo 
(Theodore F. Morse) 1907
The Japanese Sandman (Richard A. Whiting, Raymond B. Egan)  1920
Japansy (John Klenner, Alfred Bryan) 1928
A Japanese Dream  (Jimmy McHugh, Dorothy Fields)
Japanese Breakdown  [Scottdale String Band]  1930

On the Historic American Sheet Music site at Duke University, one can read the lyrics for:
Japanese Lullaby (Reginald Decoven, Eugene Field) 1890
Poppy-Time in Old Japan (Dulmage and Myers) 1915
Poor Butterfly (Raymond Hubbell, John Golden) 1916
My Yokohama Girl "In your pajama Run and tell Mamma... I will give some simolian To some Mongolian Yokohama girl like you" (Alfred Bryan, Harry Tierney) 1917
A Rose of Old Japan (Joseph Kiefer and Billy James) 1919

and don't forget: 
Happy Jappy Soldier Man, Japanese War Song (Paul West, John W. Bratton), 1914
Dolls of Many Climes

Sheet Music, 1914, US

Geo. Spaulding, music; Jessica Moore, lyrics

"Tales of Dolls from Many Climes Told in Melodies and Rhymes." Japanese doll: 

Now many years ago, 
In ancient Tokio, 
There lived a little Japanese doll. 
She had a funny name, 
Which no-one could declaim, 
And so she was called just Pretty Poll. 
Her very slanting eyes, 
You may at least surmise, 
Made mischief for herself and those those who'd call. 
And to them all she'd say, 
In her peculiar way, 
Just nothing, nothing at all.
My Little Kokomo

Sheet Music, 1905?, US

Shephard Camp

Nice image of a doll on cover. I don't own this so I don't know what the lyrics are like--presumably Kokomo is a Japanese girl.
Japanese Doll Dance

Sheet music, 1939, US

Alfred G. Robyn

No words, just the music, but interesting for the period. 
Part of a series, Dances of the midgets, monkeys, sunbeams, sea nymphs etc. for kids.

 The Christmas Toys Wake Up

Sheet Music, 1926, US

Florence R. Kirk, author; music by John V. Pearsall

An operetta for children; one of the characters is a Japanese doll, Miss Peach Blossom, who arrives on a rocking horse: 
"Honorable Santa Claus, ohayo! Honorable horse come with much slowness to honorable party.... Oh, my honorable Japan! If only I could sail back on honorable fan!" 

She sings a rather ordinary lullaby about the horse.