The airplane follows an arc in the turn because of the vertical fin. When you bank, it introduces a lateral vector to the wing's lift force. The lateral vector makes it displace sideways as it's moving forward. Without the fin to provide a weathervaning effect, it would simply continue more or less pointed on its original heading, but on a slewed track, so that the airflow is striking the body from ahead and to one side.
Think of a helicopter hover taxiing, moving forward very slowly. You apply left cyclic to tilt the rotor disc a bit as you move forward. The machine continues pointing where it was pointing, but starts to move laterally because the rotor's lift vector was tilted. If it was pointing North, it would continue pointing North, but actually proceed in a straight line Northwest.
It won't actually start to arc around with a heading change until the lateral flow hitting the tiny little fin at the back starts to point it if you're going fast enough, or you move the tail yourself with some pedal input. When you bank the airplane, the same forces are at work.
On the plane, the offset flow causes the fin (the fixed fin, not the rudder) to generate a local lateral lift force, a weathervane effect, realigning the body into the flow. The slewing or side slipping action followed almost immediately by a realignment by the fin results in a turning arc.
It's desirable to have a slight lag in this activity, because this initial side slipping action is required for dihedral effect to work, to provide lateral stability. Dihedral effect requires that a little bit of lateral sideslip be allowed to develop when a bank is induced, to create the differential lift that tends to restore the plane back to level flight, before the fin starts to do its realignment function. Fin sizing has to take this into account; too big and the plane starts to align its yaw axis to the flow instantly when a bank occurs and you end up with a spiral tendency, and too small, the plane slithers and slides around.
Note that it's the fixed fin, not the rudder doing this. If you have your feet on the floor, the rudder just trails in the airflow, not doing anything. If the plane has no adverse yaw, you can leave your feet on the floor while maneuvering and the plane will maintain coordinated flight because the fin itself is able to continuously weathervane the body into the flow.
The C of G is simply the point at which any rotational movements are centered. Pitch forces come into it mostly to compensate for the loss of the purely vertical component of lift that was reduced when the bank was induced, to keep from descending. Elevator inputs change the trim force balance to increase the AOA that will be sought by the plane's static pitch stability forces.
Since increasing AOA increases the lift vector perpendicular to the body, the increase raises both the vertical and lateral lift forces, increasing both the lift required to keep from descending, and the turn rate resulting from the lateral lift vector. So turn rate is influenced by both bank angle and elevator input, because both factors influence the lateral component that wants to displace the airplane sideways, with the pitch factor only a minor part of it until the bank angle gets steep. As the bank steepens, more and more of the total turn rate is coming from pitch rate from the elevator (take it to the extreme and imagine a turn at near 90 degrees bank - nearly all the change in heading is from pitch since you are basically flying a loop laid flat).
While all that is going on, the fin just sits there keeping the body aligned into the flow, with the rudder chipping in to help out via your feet to compensate for adverse yaw or bumps, as required.