Nick- Most performance based programs have an audition requirement. The top programs at the "name" schools are highly competitive. Another factor you mention is solid academics and the ability to double major: programs like Juilliard/Columbia & Tufts/NEC tend to admit VERY few annually, yet there are a number of programs that support and allow double majors, such as Oberlin. Bard has a mandatory double major requirement. Most of this will require 5 rather than 4 years.
I'd suggest reading Bassdad's post under featured discussions, as well as the audition tips and info thread. Then take a look at the Suggestions of Colleges with Music Major thread pinned to the top of the board.
That will give you a lot of background.
Using the Search This Forum feature, I found these jazz specific threads. There are more.
Downbeat anyone? music major (drum set) Help me please, I am pathless... Schools with strong jazz programs as well as pre-med. http://mfa.loyno.edu/thelonious_monk/index.html
You can also search for double major threads. There have been numerous past discussions here, across many schools and disciplines.
Sort through some of this info. I'm sure you'll have many questions.