hi, I am a master mason and i have lived in qatar , i worked in camp asyliyah and al udeid air force base for a couple of years, like you i also wanted to become a master mason at the prince hall lodge in al udeid but did not get a chance because i got deployed to iraq
as far as you being a muslim, that would not be a problem , masonry accepts petitioners of all faith's , but the fact that you are a qatari might be a problem, because the lodge is located inside the us base , unless you have regular access or you work inside the base,
another thing you might want to check before joining are the laws in qatar regarding masonry, because its a fact that a majority of middle east countries has a ban on masonry, that is why there are no lodges in doha except for the one inside the base
as far as going to the states and attending any lodge there, the answer is no, there are some lodges that you cannot attend and wont recognize you
but you will definitely be able to attend or be recognized in all PHA lodges
try to contact brothers from the base and find out more, dont be afraid to ask a mason