In Russian there's a very specific word for this (родина), which has equivalents in French (patrie) and in Spanish (patria). Alas, in English we don't have a term that would just convey it as eloquently, so instead we use "motherland", "native country", etc. Those do not convey the exact meaning of the word "patria" however, in my opinion.
My grand daughter was born in Luxembourg and lived there a very short time before her parents moved to Normandy (her mother is from that region, her father British). Now 20, she considers that her patrie is France, but her country of origin (ie birth country) is Luxembourg. It's a matter of education, culture and personal preference.
No. Your home country is where your home is now. You may even have a passport for that country. Your country of origin could be on the other side of the planet. A Bangladeshi who emigrated to the UK as a child and is visiting the USA and is asked where he comes from would say that his country or origin, his birth country, is Bangladesh, but his home country is now the UK.
"Patria" would be the subject so it should be "es". Here is a link to wordreference saying "Los Estados Unidos tiene" so EEUU should be singular too. http://www.wordreference.com/es/en/translation.asp?spen=Estados+Unidos