So, far, it seems that "Ida Red" is an old fiddle tune to which have been applied verses from numerous sources.
The verses belong to previously existing songs, so Randolph appears to be correct; it is a fiddle tune with "idlesome words." (Haven't found the house interior lines (lamp on the table, etc.) yet.

References to its existence during the Civil War seem to refer to some of the floating verses, some of which have minstrel origins. The story about a man named Ida Red? There is no supporting evidence, so far.

There is a parallel fiddle tune called "Chicken in the Bread Tray" or "Granny, Will Your Dog Bite?" Here is what Talley says about it. "Ira Ford, in his Traditional Music of America (1940) lists this as a square dance tune, with the lyrics as "occasional verses" fiddlers sang in calling sets. It has been called [Chicken- Granny] and has been collected widely from Mississippi to California [Note- also by Brown in North Carolina and by Randolph in Missouri]." "Ray Browne, in The Alabama Folk Lyric [1979], notes that he has heard it often as a banjo tune and that it "seems to be a greater favorite with Negros than Whites." The opening quatrain appears often in white old-time music recordings of the 1920s." Lines appear in "Shootin' Creek - Ida Red."

CHICKEN IN THE BREAD TRAY

Auntie, will your dog bite? -
No, chile, no!
Chicken in de bread tray
A makin' up dough.

Auntie, will your broom hit?
Yes, chile! Pop!
Chicken in de bread tray,
Flop! Flop! Flop!

Auntie, will yo' oven bake? -
Yes, jes' fry!-
What's dat chicken good fer?-
Pie! Pie! Pie!

Auntie is yo' pie good? -
Good as you could 'spec.
Chicken in de bread tray;
Peck! Peck! Peck!

Talley, Negro Folk Rhymes. 1991 new edition ed. Ch. K. Wolfe, p. 6, Univ. Tenn. Press. Don't have any collection dates for it.