I'm heading off to Copenhagen in second semester next year for six months. You have to apply essentially a year beforehand, so you can't go on exchange until your second year at the very earliest. You're limited in some respects to what unis you can go on exchange to, especially for law - apparently they'll only let you do law at unis with a good reputation for law, and you can't do it at most american unis because it's a postgrad study. Similarly with business at most american unis.
I think you can go on exchange to UPenn, which is Ivy League. Other decent american unis are all of the UC ones (Berkeley, UCLA, etc). However, there's a lot of competition getting in to these unis - Berkeley in particular has a shitload of applicants, and only the very best get in. Other popular unis include whatever the main one is in Paris, and Dublin.
Living expenses can be very expensive - I reckon I'll need more than $12,000 for flights and living expenses just for six months in Copenhagen. You get a small grant from the Vice-Chancellor, but for the most part you have to supply the money yourself.
And yes, you do get credited with a semester's study towards your degree.