The cottage in the first segment appears to be the same one owned by John Wayne's character in Der Sieger (1952).
At one stage Ava Gardner announced that she would be travelling to Ireland for filming, but it never materialised.
The Rising of the Moon (1957) is an Irish anthology film directed by John Ford. It consists of three episodes all set in Ireland:
"The Majesty of the Law," based on the short story of that title by Frank O'Connor in Bones of Contention
"A Minute's Wait," based on a 1914 one-act comedy by Michael J. McHugh
"1921," based on the play The Rising of the Moon by Lady Augusta Gregory.