If it helps at all, I was born in Canada. Became a US citizen, then applied for Irish citizenship. I sent my application to the Irish consulate in San Francisco. I proved Irish lineage through a grandparent. Got this very cool certificate back from the San Francisco consulate, along with all the supporting documents I had sent with the application.
Yes. I was born in London, England to Irish parents. I am now a US citizen and I have one.
Yes, yes and yes. Just know that if you get a citizenship with a pre-existing US citizenship the rules state that you have to relinquish US citizenship. Fortunately this is not regulated. Don't give them a reason/don't cause trouble.
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