You've mentioned vertical a few times, but which vertical do you mean? Are we talking relative to aeroplane axes, perpendicular to prop shaft or vertical in relation to the ground reference? The first one is very much dependant on engine installation, as that is not always completely in line with the aeroplane's horizontal axis, the second one would be mostly irrelevant for situations where local airflow is lined up with the prop axis, but becomes interesting when you start moving the aeroplane to disturb this relationship. As for the third one, with all the different attitudes possible within even a normal category aeroplane it wouldn't be very useful to start on discussing this.
When talking about loadings on a prop disc in a turn, I'm not sure you can say that the wing influences the prop in any way. The wing influences the balance of forces on the entire aeroplane, which in turn influences the aerodynamic loading on the prop and it is this aerodynamic loading, together with gyroscopic forces and the torque of the engine that mostly determines the forces within the prop. P-factor is one way of modelling this aerodynamic loading due to AoA (could be used for side-slip as well). Torque is an input from the engine to the prop. Rotational slipstream is an effect from the rotationg prop, affecting the rest of the airframe. I wouldn't describe this as an influence on prop loading.