First we would start with defining what you mean by "Islamic Countries". In this I assume you mean countries where the majority of people living in them are Muslims. However, if you were to ask Muslims how many countries today are governed "Islamicly" you would get most of them saying none and a few possibly naming a couple.
I can say I have never met any Muslim who has any interest in limiting Human Rights for all.
In terms of a comparison this is difficult for me as I have only spent time in Egypt and Turkey. You have such long cultural history in these places that I don't think you can compare the way U.S. culture deals with one specific issue to the way they may handle it there and get much benefit from it.
If you look at the field of Cultural Anthropology they do their research usually based on the researcher going and living and interacting with the culture being studied for at least two years. Why is this important? Because without it you don't understand what the practices and dealings mean. Almost always you will take them out of context because you will be putting them in to your own context based on the culture you live in.
Some here brought up Turkey. In Turkey "secular" means something different than it does here in the U.S. when we use the term. There it means actively against religion and religious people. Here it means no stance. An attempt to stay neutral. In Turkey the Friday Kutba (Sermon) is given to them from the State Authority. The same state authority that also outlaws ladies from wearing a head shall in any state college and forbids teaching reading of the Qur'an in Arabic to anyone under 14 years of age. So its all over the place.
How they got to this point is complex, the history and what they mean for the people living there is important to consider.
To answer your core question the leaders of Progressive causes in majority Muslim countries would be Indonesia, Turkey and Egypt. Before one says that is crazy to call any of them Progressive, simply look back at the United States 50-60 years ago. We have changed dramatically since then. You'll see a lot of change out of the so called "Islamic" world in the near future.
In my opinion the Progressive principles you raise are more a product of ancient cultures than are codified anywhere in the teachings of Islam. There is no verse in the Qur'an dealing with homosexuality. Man is the one obsessed with sex, not God.
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