No it's not. Ulster is in Ireland, Liverpool is not in London.
I'm from Ulster and have lived there all my life until very recently. "Paddy" is a derogatory term used - even by some Ulstermen (though not all) - for people exclusively from the Republic of Ireland/
Nope, you'll find that in England it is used to describe people from anywhere in Ireland.
And part of ulster is in the republic ?!
I believe the technical term is 'arrogant little ********'.