I think most of the members were actually English, weren't they? Obviously Shane MacGowan was Irish though.
Also, I prefer If I Should Fall From Grace With God
They were all Anglo-Irish; i.e. part of the Irish diaspora (hence the band forming in London - that's where they were all living). However they were all Irish to begin with, or at least the progeny of Irish parents.
As for albums, does anyone else absolutely love
Red Roses for Me? I mean, I love
Rum, Sodomy and the Lash and
If I Should Fall From Grace With God too, but
Red Roses seems to be a bit of an overlooked classic in my mind. It's certainly their punkiest album.