Do the articles of incorporation and/or bylaws need to mention how the board of directors is selected, of can it be in another document, or can the nonprofit organization not have a board of directors? Can the board of directors be called something else, Executive Board for example?
Yes, Virginia nonstock corporations are required to have a board of directors...unless the members or directors expressly agree to eliminate it.
Requirement for and duties of board of directors (§ 13.1-853)
A. Except as provided in an agreement authorized by § 13.1-852.1, each corporation shall have a board of directors.
B. All corporate powers shall be exercised by or under the authority of, and the business and affairs of the corporation managed under the direction of, its board of directors, subject to any limitation set forth in the articles of incorporation or in an agreement authorized by § 13.1-852.1.
Your director selection procedure is to be set forth in the articles of incorporation. The requirement exists not only to explain how the initial board will be selected, but how subsequent boards will be selected.
For example, you'll want to determine whether there'll be members. If so, will the members elect the directors? Will the board be self-perpetuating? Will you have and ex-officio director? There's a good FAQ on the process here.
Big picture: Virginia wants companies to have a procedure that doesn't depend on specific, named people who may or may not be around in the future; it wants companies to develop a process.
As to the naming: for the state's purposes, you're creating a "board of directors," which is what you'll call it in the incorporation documents. It's up to you to decide whether it'll also have a non-legal title for day-to-day operations or development/marketing purposes.
Side note: it looks like you've already found Virginia's Nonstock Corporation Act, § 13.1-801. If you get into a pinch, I've always found the State Corporation Commission to be helpful by phone.