I don't know much about the historical context, but it sounds to me like the people were able to take elements from both cultures. In a way, they are able to go back and forth between the cultures like someone who is ambidextrous could go back and forth between using either hand. A synonym could be "flexibility."

Nice text, I think the author wanted to say that these states upon being built on the remains of different cultures that were strange to Arab and Islamic culture, that was reflected positively on building these states and made them multicultural ones.

From my interpretation, I believe it means that the Umayyad and the Abbasid and the places that were formerly ruled, have adopted each other’s cultures and are able to , for a lack of a better word, practice the lose cultures as well as their own.

Wow. I’m guessing the books you read must be for very advanced English speakers, because I had trouble understanding everything, even though I’ve lived in the UK for 11 years since I was 10. But that goes to show that if a person lives in the UK and doesn’t put the effort into learning English, his English will not improve, no matter what country he’s living in.