On the question of if charters are public...

...the National Labor Relations Board has just decided "no":

In its recent decisions, both issued Aug. 24, the National Labor Relations Board ruled that Hyde Leadership Charter School in Brooklyn and the Pennsylvania Virtual Charter School are — like other government contractors — private corporations that receive taxpayer dollars. 

In the New York case, for example, the board found that even though state law describes charter schools as existing “within the public school system,” the schools were not directly established by a government entity and the people who administer them are not accountable to public officials or to voters. “Hyde was not established by a state or local government, and is not itself a public school,” reads the board’s majority opinion, signed by Democrats Kent Hirozawa and Lauren McFerran. 

The decisions mean that the schools’ employees must organize under the National Labor Relations Act, which applies to private-sector employees, rather than under state laws that apply to public-sector employees.