You must have a a greencard (lawful permanent resident status) first to be able to apply for U.S. Citizenship. To correct some of your statements:
One can be a refugee from a rich country as long as they can prove that they would or have suffered persecution based on their religion, political opinion, race, nationality or membership in a social group.
No, Joining the military will not get you a greencard nor U.S. citizenship and you cannot join the military unless you have at least a greencard anyway.
Its not that you would have ties to U.S. islands/territories, you'd have to be born there, just like being born in the U.S.
Generally Foreign Students on F-1 Visas, go to OPT, and then try to find a company to sponsor them for an H1-B work visa, and later try to get the same or another company to sponsor them for a greencard and then after 5 years apply for U.S. Citizenship.
Speak with an immigration attorney about your options.
Andre Olivie, Esq.
Legal disclaimer: This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Do not rely on this advice without speaking to an immigration attorney in detail about your case. This message does not create an attorney-client relationship.
You cannot become a US citizen without first having the green card.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship.