I am doing three away rotations this year. None of them charged me anything, although one of them made me jump through the most ridiculous hoops in order to apply (extra immunizations on top of the ones required by my school, pages and pages of BS paperwork). One of them even paid me a very generous stipend for my month there. It covered my housing, and then some.
Occasionally you can stay with residents or other med students for very little or free, but you should contact the school early to make arrangements for housing, as sometimes you will have to fend for yourself, as someone mentioned earlier. On one rotation, I even managed to get paid to house-sit for one of the residents.
As far as your white coat with your school's name....it has been my experience that students are encouraged to wear their own coats complete with school name and/or nametag. That way, people can easily identify you as a visiting student. But then, I have been doing pathology rotations, and after awhile I realized that most places didn't care if I wore my coat at all!