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...:)