A:
The rules for small boards such as you describe are relaxed in comparison to those for general meetings (see RONR pp. 477-78).
Among these is the rule that "subject to rule or custom within the particular board," the chairman can make motions and vote publicly.
If, by long continuance of the practice of disallowing the chair to make motions or vote publicly, your board has established a "custom," then the board may adopt a rule (by two-thirds vote) that allows him these rights.
It might be worded something like
To make a motion, the chair would say something like
"The board's chairman may make motions and vote publicly on all questions for which he has no direct personal or pecuniary interest not common to other members of the organization."
The qualifier about personal or pecuniary interest is true for all members of the organization (RONR p. 402).
It is spelled out in this rule only because, if it were left out, he would be exempt.
Note that "the board's chairman" refers to whomever is conducting the meeting, not necessarily the Chairman of the Board.
No motion in a small board needs a second.
If the motion is debatable, he could immediately speak on the issue, and then ask "Is there further discussion?"
Debate would continue as usual, followed by the motion's disposition (usually a vote on adoption, but possibly a vote to postpone it, send it to committee, etc.)