You raise a lot of issues here, most of them related to state law or your bylaws.  I am not aware of any state nonprofit corporation law that makes it illegal for a paid executive to serve as a member of the board.  But admission to the board would be controlled by your bylaws and it does not sound as though this appointment went through a standard selection process.  Assuming the directorship was validly conferred, a few states may have limits on a paid executive “controlling” a board, but that is not the general rule and it is not clear whether the exec has a position that gives her legal control over the board. 

Who has authority to agree to the salary is an internal question for the organization?  If whoever set the salary had no authority to do it, the full board may have power to terminate the arrangement or change the salary if it is deemed too high (even if not an excess benefit).  A dissenting director (or member if you have voting members) could conceivably sue whoever was responsible to recover damages for the corporation resulting from the overpayment, but that is unlikely to be economically worthwhile to try to do so. 

The resignation of directors because of internal strife like this is not likely to be grounds for cancellation of your D&O insurance but you may have to report some of the strife in order to obtain an effective renewal.  You should also be aware that litigation over issues like you suggest here may not be covered by your D&O insurance under the frequent exclusion for claims brought by one named insured against another named insured.  The officers and directors are all named insureds.