In most places, yes. Sexuality and marital status for the most part do not affect Immigration status.
Additional Information:The question also seems to ask whether the immigrant can seek legal residency or even naturalization by virtue of his or her marriage to a citizen of the same sex. The answer to that question is that, generally, countries which have nationwide recognition of either same-sex marriage or civil partnership will accept an application from the spouse of a citizen, regardless of the genders involved. However, the United States is not one of those countries. Although the official policy of the United States is to immediately dismiss all applications based on a same-sex marriage, lately such cases have been held "in abeyance" and deportation orders have been adjourned in light of questions concerning the constitutionality of the law prohibiting the recognition of same-sex marriages. To my knowledge, though, no one has been granted permanent residency or naturalization on the basis of same-sex marriage to an American citizen.I don't think you can. You either has to be single or married to adopt. Just living together is neither. if you are in a samesex relationship and has gone through partnership might be possible though.
Anyone can do anything they want to as long as they truly want it. No, in Islam having a boyfriend/girlfriend type of relationship is not allowed whether your a male or a female and whether the bf/gf is Muslim or not. If you have anymore questions regarding the matter, just ask and I'll be happy to answer them. i gree in islam having a boyfrnd or a grlfrnd is not allowd but today no youngr generation follows it .....so if a muslim girl personlly like a non muslim boy...then it is not allowd in islam.....u cannot marry him unless he accpts islam...:)