Regarding the disturbing PEW study the OP pointed out. I found a relevant article, Bill Maher and Sam Harris’ proof is wrong: Their argument is based on an untrustworthy poll, where it brings up an important point:
Am I saying that Maher and Harris are right in asserting that Muslims in general—or in other words, in their majority—hold savagely violent, intolerant and misogynistic views? That would be the case if I trusted the Pew poll. But I don’t. What I am questioning here is not the methodology of the respected research Institute, but rather the genuineness of the answers provided by many of the 38,000 individuals it surveyed. Just picture the typical polling interview. Imagine you live in a country where Islam is the religion of the State, where criticizing the religion (let alone leaving it) is a criminal offense, where the educational system and the pervasive state media gang up every day to hammer that Islam is the highest moral norm ever—where, hell, even the opposition (mostly made of Islamist groups) does nothing but double down on religious intransigence… And here comes the Pew pollster, a total stranger with a list of disturbing questions pertaining to religion—questions to which the wrong answers can get you in trouble in many ways… Not the best conditions to conduct a credible opinion poll.
Polls, of course, are anonymous, and trained pollsters have their ways to fabricate empathy, including asking the interviewees if they had a good day before engaging them on more serious matters. But as nice as a warming-up chitchat might be, it simply cannot fend off the self-preservation instinct built over a lifetime’s experience of psychological pressure. Of course, the extent of the pressure (which itself depends on the extent to which religion is used as a political tool for social control) varies from one Muslim country to another. But then, so do the answers. While the Pew numbers indicate a high prevalence of the opinion that Sharia should be the law of the land in countries where Islam is the religion of the State (91 percent in Iraq, 83 percent in Morocco, 74 percent in Egypt…), the rate drops to 12 percent in Turkey, where secularism is a forefront constitutional principle. In other words, the more the questioned citizens are coerced into religiosity, the more likely they are to pick the safest answers—those consistent with what they were force-fed about religion since they were kids—when a pollster comes around.
So are Muslims in the US and other secularized countries lying about their disapproval of Sharia? Or that the Muslims under Sharia are lying about their support of it? Which seems more plausible? I'm inclined to think the latter.