Getting back to the original question, my understanding is in alignment with Fred's.
A Closed unit may select "additional" criteria which must be satisfied by it's membership (i.e. member of x church, resident of the sponsoring HOA, etc.). It is up to the BSA and the Chartering Org to determine what additional criteria are allowed or not (i.e exclusion based on race is probably not permitted) prior to the BSA (well local council) issuing a charter.
As most of the rest of this discussion seems to wrap around, a closed unit may not change the other mandatory BSA membership requirements.
As for a Charter School sponsoring? It may vary state to state, but my belief is that if the charter school has a separate and distinct tax-id from that of the School District, then it is a separate legal entity (corporation, partnership, etc.), and may enter into any additionally permitted business operations - so that in addition to running the charter school, they would be able to sponsor a scout unit.
The best way to describe this is that the charter school corporation is then using it's "profits" to provide a charitable contribution in sponsoring the Scout unit. Even a non-profit charter school could do this as the non-profit status does not mean not making money, it means that any excess funds are used for public good and not to enrich shareholders.
The last issue then is the facility itself. If the Charter school (corporation) owns/leases it's own facilities, then it would be free to use those facilities as it sees fit (per the lease agreement if any). If the charter organization is getting "rent-free" use of the pubic school facilities (i.e. maybe running a school in a school or similar), then the issue becomes a little more complex.
If the (public school) facility allows other groups to use the facilities, then those same rules would probably govern the charter school sponsored unit's use of those facilities.
Edited by gumbymaster, 09 February 2017 - 01:24 PM.