Cowboy's Wild Song to His Herd: The poems in Rhymes from the Rangeland are Beggs's own (see the introduction/"explanation"), so he may be considered the original author of the text of this song. Lacking any indications that it was turned into a song prior to the Carters, we must assume they supplied the music. Per Charles Wolfe, sources differ on whether A.P. or Maybelle wrote the song, but considering that Maybelle was by far the more musical of the two, I strongly suspect the tune at least was hers.

Richie, thank you for hunting up the original poem! I'd thought of doing my own rework, but the Carter attribution put me off. And "Down by the Garden Wall" (Darling Daisies) was a great find as well.

Darling Nellie Across the Sea: Per Charles Wolfe, Sara wrote much of the song herself (implying an antecedent).

Funny When You Feel That Way: Per Wolfe, 19th c. versions credit G.W. Hunt.

The Girl on the Greenbrier Shore: Per Wolfe, it's based on an incident which occurred in 1896.

He Never Came Back: Wolfe gives the original date as 1891, not 1892. The sheet music at the Levy may not be from the first publication run.

Jealous Hearted Me: The Carters learned it from Lesley Riddle.

My Dixie Darling: Per Wolfe, the original song was titled "Dixie Darlings" (plural) and it was written in 1907, not 1909.

My Native Home: Per Wolfe, by John Rogers Thomas, c.1865. Learned from a woman in Russell County, VA.
The Levy site has "My Dear, My Native Home", music by John Rogers Thomas, words by Charles Hart, 1855. Same song? I haven't really checked.

My Old Cottage Home: Per Wolfe, written by R.A. Glenn (with two n's).

One Little Word: See thread 111888; sadly it didn't show up when I ran a search on the title just now. The thread contains some source links and a transcription of the original lyrics (supposedly). Note: Frank Howard's "One Little Word", available at the American Memory site, is an unrelated piece.

Over the Garden Wall: See thread 6020, where I've provided the original lyrics and an ABC of the original melody.

The Spirit of Love Watches over Me: Per the sheet music linked above, the original title was "Thou Art Gone from My Gaze".

Two Sweethearts: Per Wolfe, "Moran and Helf" were E.P. Morgan (words) and J. Fred Helf (music).

When the Roses Bloom in Dixieland: Wolfe gives 1913, not 1915, as the date for Evans' original. George "Honeyboy" Evans was a minstrel showman. He also wrote "In the Good Old Summertime".

I'm skeptical when people talk about a song as having been written by A.P. Almost all the Carter songs were copyrighted in A.P.'s name only, regardless of which of the Carters were actually involved in writing/reworking them. A.P. doesn't appear to have been musically accomplished--his bass lines are glaringly dull and often at odds harmonically, and he needed Riddle to accompany him on the song collecting trips to capture the tunes they heard. I can't recall reading that he ever played an instrument. So I suspect that Sara, Maybelle and Lesley Riddle did nearly all of the music writing and arranging, and much more of the lyric work than they're generally credited with.