1. Yes, absolutely. Proving you can cut it in the classroom is very different from proving you can do it in a work setting.
2. My first internship was actually at the university I was studying at. I stayed over the summer and helped one of the professors with some fuel cell work. As it turns out, that work and the summary paper I wrote about it helped me get my current full-time position. You're just starting out, so anything relevant you can add to your resume is helpful. I'm not saying you shouldn't try for the NASA one you mentioned or anything like that, but don't count out something "smaller" for your first. Anything can help, it's never really going to hurt you to get experience and some networking contacts.