"Liberals" in the US - democrats - are indeed center-right of the rest of the world. Look at Obama, Clinton, Biden. They are very center on some issues and quite right on others.
Probably. I've always considered both US parties to be so far right wrt the rest of the world, that anything even remotely moderate would be labeled "liberal" or "communist". Both terms used with extreme prejudice and disdain, of course.
A U.S. "liberal" is very socially-progressive (pro-gay marriage, pro-choice, pro-environment, anti-racist, mostly pro-regulation and anti-corporate). I think that's the sort of people the parent poster intended to describe. In Europe "liberals" are usually pro-business and socially-conservative.
(Btw, I wouldn't say a U.S. liberal will automatically sit on the right of the European discourse, today. Traditional socialism has virtually disappeared as a political choice in Europe as well, so really there is very little disagreement today between a U.S. liberal and a European with mainstream social-democratic sensibilities -- except maybe on foreign policy.)
There is no equivalent of a European left in mainstream US politics. You see bits and pieces in some small-time candidates like Bernie Sanders, but nothing serious. The red scare did its job.