Answer the first question, then I would tell you the rest... Are you or do you own a legal education establishment (colleges, Universities, etc) ?
For a Edu domain , you need to have apply on official letterpad of the college, university to the Edu Registry. As soon as they verify you will get an Edu domain. It is not open for general public..
You need to establish an educational institute in order to register for a .edu domain. You won't just get it. It has to be reviewed by ICANN.
Check here the eligibility criteria to register .edu domain: http://net.educause.edu/edudomain/eligibility.asp
I think .edu and .gov domains are not in elite class as earlier . Matt cutts explaind Google's policy about .edu and .gov domains , they are considerd same as .com and .org domains .
All domain extensions are considered equal. Its the trustrank that matters. of course, .edu or .gov would be used for reputable and legitimate course and not something "unethical". Thus the SE's put a factor of trust on these extensions. Apart from that, all domain extensions are considered equal, at least in my opinion.
here is what I have learned, maybe it will help someone else… You have to purchase the domains through EDUCAUSE (educause.edu) In order to purchase, you must be a postsecondary US institution that is accredited by one of the agencies on the U.S. Department of Education’s list of Nationally Recognized Accrediting Agencies
you have all the papers needed and you still dont know who does the registration? when there is only 1 place to go? BTW, there are no papers needed, just fill out the online application, then the verification process begins, shortly after “your application was denied†email arrives and you're back to square 1. PS. Hope you won't break over it.. Great post, made my day , thanks!