For Delta Chi: advisory positions are open to non-members, regardless of gender. The only position a non-member cannot hold is "BB" which is the primary chapter advisor. (He typically attends all or most chapter meetings and submits reports to HQ every semester) The majority of the Alumni Board of Trustees (which the "BB" is part of) must be made of Delta Chi Alumni, but I can think of a few chapters that have women, and non-Delta Chi men on their ABTs.
Delta Chi does not have honorary membership. For you 50 year-old woman example the answer is no. Alumnus Initiation, as the name suggests, is only for men and Initiation is a private ceremony. Not directly related to your questions but our Associate Member Ceremony is open, and female and non-Delta Chi advisors can and do attend.
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"Delta Chi is not a weekend or once-a-year affair but a lifelong opportunity and privilege"
- Albert Sullard Barnes
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