Here is what Robert's Rules says about ex-officio members:
51. Ex-Officio Members of Boards and Committees. Frequently boards and committees contain some members who are members by virtue of their office, and, therefore, are termed ex-officio members. When such a member ceases to hold the office his membership of the board terminates automatically. If the ex-officio member is under the control of the society, there is no distinction between him and the other members except where the president is ex-officio member of all committees, in which case it is evidently the intention to permit, not to require, him to act as a member of the various committees, and therefore in counting a quorum he should not be counted as a member. The president is not a member of any committee except by virtue of a special rule, unless he is so appointed by the assembly. If the ex-officio member is not under the authority of the society, he has all the privileges, including the right to vote, but none of the obligations of membership; as when the governor of a state is, ex-officio, a manager or a trustee of a private academy.
If your bylaws don't specifically say if the principal has a vote or not, I believe the above passage would say to allow it. If you don't think the principal should have a vote you would need to ammend your bylaws. For example, mine says:
Section 5 - Voting Privileges
All members of the Board, with the exception of the principal, may cast one vote. No vote may be cast by proxy. If the chairman of a committee is not present to cast a vote, one person representing that committee may cast one vote.
As for teachers being officers, if your bylaws allow all members to be eligble for officer positions and then define teachers as members then they should be able to run. My group's bylaws define members as:
Section 1 – Definition
The membership shall consist of residents within the district's schools and of parents/guardians of students attending the district's schools as well as the faculty/staff of the district's schools.
But define officer candidate eligibility as:
Section 2 – Candidacy
Eligible candidates for elected or appointed positions shall be defined as those who have a child/ward attending District 51 schools or faculty employed by District 51.