Just adding comments to the overall reasonable list above.
Princeton doesn't have sororities.
Stanford fits criteria, except for difficulty in admissions (i.e. definitely one of the five hardest in the country).
Dartmouth having "reasonable weather" depends on your definition of reasonable. It can get pretty cold up there in NH. It's preppy too (like Duke).
Williams doesn't have sororities (or school spirit really).
Amherst doesn't have sororities (or school spirit really).
UPenn fits criteria assuming you're not from Florida and Philly's weather is ok for you.
Georgetown doesn't have recognized sororities (although they might have some "underground" ones, I'm not sure).
Rice doesn't have sororities.
UVA fits your criteria.
UNC fits your criteria.
UMich fits your criteria, although A^2 can get brutal in the winters.
UC-B, UC-LA, USC (I don't know anything about UC schools)
Tufts doesn't have school spirit (does have sororities, though). Weather is subjective.
Northwestern fits your criteria (except, again, weather in Evanston is debatable).
Vanderbilt fits your criteria (super southern preppy too).
Emory fits your criteria, although school spirit not as good as at places like UVa, UNC, Vandy, Northwestern, UMich (i.e. ANY school in DI sports).
I think Duke's biggest overlaps are UPenn, Dartmouth, Cornell, UNC, UVa....