"Old Yeller Dog" probably refers to voters in the South who voted solely for candidates who represented the Democratic Party. The term originated in the late 19th century as a reaction against the Republican party of Lincoln (and abolition). These voters would allegedly "vote for a yellow dog before they would vote for any Republican".
The tune/song is a variant of an old blackface minstrel song called "Down in Alabam" or "Aint I Glad I Got Out De Wilderness", more familiar nowadays as "The Old Grey Mare" (She Ain't what She Used to Be). One version of the song was used as a campaign song for Abraham Lincoln ("Old Abe Lincoln came tearing out the wildnerness").
The banjo tablature is by John Letscher.
The presence of dogs in a house of worship may seem incongruous to us now, but the practice was not rare at one time, particularly in the South. South Carolina minister Charles Woodmason found it necessary to bar his congregation from bringing their animals with them to church in the late 18th century. Not only were they troublesome, he explained, they were also "an affront to the Divine Presence...to mix unclean things with our service." The congregation was only imitating longstanding practice in Britain. In James Hall's Travels in Scotland, by an Unusual Route: With a Trip to the Orkneys and Hebrides (1807) he states:
"I was amazed to see how much the ministers in the interior of the Highlands are plagued with dogs in their churches. As almost every family has a dog, and some two, and as these dogs generally go with the people to church; so many dogs being collected often fight, and make such a noise during public worship, as not only disturbs the congregation, but endangers the limbs of many."