While you can do any study abroad program offered by your school, I am not aware of any study-abroad aviation programs. They may be there, I'm just not aware of any. Also, there are likely to be more study-abroad opportunities at more rounded schools (Purdue, UIUC) than at schools that are focused on aviation.
For the second question, most students at Purdue (I can't speak for other schools) do not fly at Purdue over the summer, but most don't stop flying all together. As soon as you have a private pilot certificate, you can rent planes from local airports anywhere in the country, so a lot of students rent planes at home over the summer. Also, after you earn a commercial pilot certificate (2nd year at Purdue - again, I can't speak for other schools) you can do commercial flying in the summer. Many students become instructors, as I did, but there are also a lot of other jobs for low-time pilots. I know of people from Purdue who have done bannertowing, glider towing, and corporate flying over the summers.
Hope that helps - let me know if you need more info.