The word امرأة (a woman) is a feminine word that ends in Ta Marbúta ة which is spelled "H" when the word is alone or at the end of the sentence but when it is in the middle of the sentence or its beginning and needs to be moved with a vowel, then H turns to T. I made some longer text if you like to read: https://forum.duolingo.com/comment/33454574
check under the word غاية where I talk about Ta Marbúta and Tanwin (Nunation)
There is an order but it's not related to the countries or anything, but mainly by the logic of things. Sticking to the question here, I guess you were asked to translate into Arabic, and the order of the adjectives in Arabic is the reverse of that in English. Because in English, the most relevant adjective comes just before the noun, while in Arabic, the most relevant adjective comes right after the noun.
The logic (let's stick to the English one here):
- Arab American woman.
- American Arab woman.
In the first instance, you are talking about an American woman who is originally an Arab. In the second instance you are talking about an Arab woman, who is originally an American.