I have a roll that looks almost recut but the label (what's left of it)
says "The Teddy Bears' Picnic - Number 03854".

I seek as much detail as possible: what company did this roll come from,
played by, words by, copyright, etc.

Duaine Hechler
http://www.hechlerpianoandorgan.com/ 

 [ The music of "The Teddy Bear's Picnic" was composed by John W. Bratton
 [ in 1907.  The words were added by Jimmy Kennedy in 1932.  Roll number
 [ 03854 might be Imperial or QRS.  -- Robbie