It comes down to optimism again I think. A European devout Christian would certainly pray for their sick child - but they would not expect their prayer to be necessarily answered, so they would also take the kid to the doctor. If the doctor cured the child, they may attribute the good luck to a form of divine intervention whose instrument was the doctor.
The problem in the US is that many local forms of Christianity have set themselves in opposition to science - either/or - and so the doctors are not instruments of God but His antagonists. I don't have a really good explanation as to how this came about.
People whose religion is sincere will inevitably come into conflict with religious and secular authorities from time to time. Over several centuries, most of them left Europe and went to the US, where there were no religious authorities, and the secular authorities were much more hands-off. It's not particularly surprising that their descendants still take religion seriously.
If we had sent all the criminals to the US and all the puritans to Australia then perhaps the results would have been switched.
It's always strange and interesting to hear people harp on one particular minority group of colonists as exemplars. Contemporary conservative religious groups have little or no connection to Puritanism at all.
From what I understand you will be quite left-out of society in most of US if you don't go to church. Here in Sweden, as a member of church, you can have funerals, marriages and baptising at church, but the church will never really bother you to come to the Sunday service.
The Swedish church was slightly decoupled from the government a number of years ago, but fees from members are still collected by the tax office, something like half a percent at most, it varies by region (diocese?).
Also, members of the Swedish church get to participate in elections for the primary positions. See http://www.svenskakyrkan.se/default.aspx?di=272326 I don't know if that happens for US churches.
Mormons and other US free-churches do come here and have head quarters. I find that they pray on the poor and unfortunate ones by knocking doors and stopping people on the streets to offer their solutions. They're very much considered fringe groups though, since the Swedish church is so much moderate and yet provide ceremonies that are in demand even in the world's most secularized country.
Not really. I don't. Maybe you're talking about people that belong to a particular church being pressured into regular attendance? But I belong to no church and even when I have in the past the only times I went regularly were when I was forced to as a child.
{edit} Maybe I've just managed to avoid areas like this but I've lived in the Detroit, Portland and Toronto (Canada, which I guess doesn't count). {/edit}