hmm i didnt know edison wanted a lot of ME... and while the choices you listed are good, those big aero companies have a certain feel/culture to them that may not be right for everyone. while you should apply if its what you want, id say to definitely open up your choices/considerations. several of my (mostly) EE classmates who are in the know and not just those people who know nothing outside of school would probably agree.
as for small vs large, id say go for both. do small one year and large the next. two very different pictures. imo a small company isn't something that is worse than a large one or simply a stepping stone and vice versa. ive worked 3 different jobs/internships or w/e u want to call them in college (finishing EE undergrad at UCI) and did one unpaid job (probably the most enjoyable one though, set own schedule too).
stories are a bit long so i will just leave them out for now. my biggest advice for an interview or getting an internship isn't to just have a high gpa b/c many have met very many high gpa students who are horrible interns and vice versa or be the stereotype charming douchebag or asshat during the interview - most intelligent managers know those people usually just full of hot gas/ego and try to talk their way through the interviews, know almost nothing, and are a pain to work with when its time. the above may work for other jobs/positions, but i don't see BSing the way through the interview work as often (unfortunately it still works too often). yes still be charming and nice, it will help, but dont be like the stereotypical sales person or w/e if its not needed. for what to do during an interview, simply put, google can word it all out better than i can with the jurable of thoughts in my mind.
my biggest mistake in my undergrad was taking a lot of best engineering courses in other majors i didnt need and doing poorly in them and dragging my gpa down a lot. also could have taken a lot of classes more seriously but i guess more for myself i forced myself to try in my hardest EE classes and did well in those.