At the moment it looks most likely that the song arrived in Ireland from America shortly before World War II via Delia Murphy or just perhaps Donagh MacDonagh.

Dry stone walls are not in themselves evidence of Irish influence; anywhere that has loose stones will have such things. I live in Yorkshire, which is full of them. The style in which they are constructed might tell you something about their origins if you look into it, though. Regional construction techniques vary quite a bit, partly depending on the materials available.