I was in your situation about 7 years ago myself, was visiting RPI, applied to MIT as my long shot. Got into MIT somehow, got my SB MechE, MS MechE (had the chance to go PhD, decided not to), working in industry now and I think I've got a little bit of perspective on the engineering school experience.
From working directly with lots of engineers, helping profs select grad students, my own job search, and helping hire engineers there's a couple things you can do that'll greatly help your success no matter where you go to undergrad. Be passionate about something, stay focused, and do substantial work on it. Whether it's Robocup, Formula SAE, rocketry club, etc, just do it and make an impact. Do design, learn, work with people, and have fun doing it. Add being friendly and reliable (don't underestimate these!) to that and you really can go anywhere.
I think I could have gotten a similar education at most reputable engineering schools, everyone has the same statics and dynamics equations, but if you're passionate about what you do a top school can be a truly exhilarating experience. Besides getting the opportunity to do substantial work during undergrad for top PIs, there was just an intense forward current that came from living, working, and keeping up with brilliant people. Make sure you've got the humility to deal with people smarter than you and can take failures in stride and you really can come out with a tremendous life experience.
Whether you do Aero+MechE, or Aero or MechE will really depend on the specific program where you eventually go. When you end up working on real projects you'll likely find the impact is small, and from what I've seen Aero and MechE resumes are often thrown into the same pile. Both programs should be teaching structures, dynamics, design, controls, thermo/fluids (different spin, but it's the same math) and you may find your time is better spent on your own independent study than the specific requirements of a double degree.
YMMV and such. Good luck!