I would like to know the responsibilities required of a small,
non-profit (501 c3) New York City based, (modern) dance company's
Executive Director

This is from the United States Dictionary of Occupational Titles:

CODE: 151.027-010
TITLE(s): CHOREOGRAPHER (amuse. & rec.) alternate titles: director,
dance

Creates and teaches original dances for ballet, musical, or revue to
be performed for stage, television, motion picture, or nightclub
production: Composes dance designed to suggest story, interpret
emotion, or enliven show, coordinating dance with music. Instructs
performers at rehearsals to achieve desired effect. May audition
performers for specific parts. May direct and stage presentation.

http://www.occupationalinfo.org/15/151027010.html

the responsibilities of a choreographer (or artistic director)
certainly may entail, auditions, instruction, rehearsal scheduling,
and stage presentation. I however am interested more specifically in
how the executive director works with the artistic director in the
business of running the dance company as a whole. thanks