That is incorrect thinking. In a two-body problem, what's usually done is to consider the reduced mass of the system, defined asBut if they orbit the C of M then surely this point has mass M+M=2M so we would have F=2GM^2/{R/2}^2?

At that point, we "have therefore formally reduced the problem of the motion of two bodies to anequivalent one-body problemin which we must determine only the motion of a 'particle' of mass in the central field described by the potential function ." - Marion and Thornton,Classical Dynamics of Particles and Systems, 4th Ed., p. 293.

The potential referred to is the potential energy corresponding to the gravitational force between the two bodies. It only depends on the distance between the two bodies.

Here's my question for you: at this point in your course, what do you know? Are you expected to derive the period from first principles like the Euler-Lagrange equation and conservation of angular momentum? Or from Newton's Second Law? Or do you start from the equations

and

?

Here, is the semimajor axis of an elliptic orbit (in your case, it's just the distance from the C of M to one of the stars), is defined by the previous equation, and is the reduced mass defined earlier.

Concerning the boat problem: so, what's going on here is that the first guy throws the snake to the other guy. As he throws it, the snake achieves a momentum to the right, forcing the thrower (and the boat that is fixed relative to him) to the left. When the catcher catches the snake, the snake then stops going right. Its momentum is arrested, which gets transferred through the catcher to the boat. The boat, you will recall, was moving left. This additional change in momentum is exactly the momentum earlier incurred. Hence, the system stops again when the catcher catches the snake. The entire problem of why the boat stops again is entirely solved by conservation of momentum considerations.

The boat-snake-man-man system acts as a whole from the point of view of an observer such that its center of mass does the same thing before and after the throw. In this case, the system is at rest before the throw. Therefore, the center of mass is going to be at rest after the throw is completed.

Does this shed more light on the matter?