A-Begging I Will Go isn't related to any Child ballads; it seems first to have appeared as The Beggars Chorus, a stage-song in Richard Brome's The Jovial Crew; perhaps not in the original production (1641), but in a revival of it, c. 1684.  (Refs. Chappell, Popular Music of the Olden Time, 1859; Simpson, The British Broadside and Its Music, 1966).

Jack Rowland is another complicated case, as Carthy wrote much of it himself, based on a narrative published by Jamieson (Illustrations of Northern Antiquities, 1814), as Child Rowland and Burd Ellen, a traditional narrative with some parts in verse, imperfectly recalled from Jamieson's childhood.  Child quotes one verse in his Fragments section (vol. V, p.201).  Various retellings of the story, rarely with proper attribution, can be found on the web.  It would be necessary to see Jamieson's text in order to establish how much is him, and how much is Carthy; although various motifs are included that occur in, for example, The Twa Magicians, Sir Cawline and The Maid Freed from the Gallows, that doesn't in itself imply any actual connection.

I'm amazed that I overlooked Long Lankin!  That was the name of a band I played with back in the '70s...