The answer to (1) is certainly yes. Although a "European-style letter" could conceivably hurt you, I recommend getting the letter anyway. People reading your application will be aware (at least somewhat) of the cultural differences. Indeed, see RoboKaren's answeranswer to a related question here.
Although I have never participated in a postdoc hire myself, my understanding is that what matters most is persuading one or two people that they really want you. If they already know you and your work, they might not read your application all that carefully, and if these people are extremely enthusiastic, the rest of the hiring committee might simply go along with it.
Another potential obstacle to keep in mind is that since you are further away, American researchers might know you less well than competing American candidates. For example, American candidates might have met these researchers at multiple conferences, and/or have close relationships with their advisors.
One way of mitigating it (which is not always possible, but it's worth investigating) is by travelling to appropriate conferences where you can meet American (or other) researchers with whom you might want to work. It is especially good if you travel with your advisor, because then he/she can introduce you and get you invited to lunch/dinner with the "bigwigs".
Good luck to you!