How to calculate optimal prop diameter, pitch, torque and power coefficient given required thrust?
How would I be able to calculate the optimal prop diameter, pitch, Kp (power coefficient), Kt (torque coefficient), as well as optimal number of blades and optimal manufacturing material? Or, would a turbofan be more optimal?



Optimal prop diameter is simple: the largest you can accommodate = 30 cm. Disk area A = 0.07 m$^2$
Optimum number of blades: as few as possible, let's start with 2.
Use ducted rotors for a more vertical alignment of the rotor tips.
The quadcopter has horizontal rotors. Tip speed should not exceed 0.8M = 270 m/s which equates to $\Omega$ = 570 rad/s = 5,520 RPM
We have a prop diameter, RPM and thrust, now we need to determine power P and torque Q to drive the prop. We'll do so using expanded momentum theory for the hover, like in this answer.




For a rotor with solidity ratio $\sigma$ = 0.1, the thrust coefficient $C_T$  is maximum 0.01, let's use a safety margin of 20% and take 0.008. Then the maximum thrust for this rotor becomes:

$$T = C_T \cdot \rho A (\Omega R)^2 = 0.008 \cdot 1.225 \cdot 0.07 \cdot 270^2 = 50N$$

We need 15 times this thrust! In order to verify the value of $C_T$ there is another graph in Leishman that relates $C_T$ with blade-pitch angle. It turns out that $\sigma$ = 0.01 is at a blade-pitch angle of about 12 degrees, we cannot go much higher than that. Neither can we increase tip speed, our only choice is disk area.



$$A = \frac{T}{ \rho \cdot C_T \cdot (\Omega R)^2} = \frac {750}{1.225 \cdot 0.008 \cdot 270^2} = 1.05 m^2$$

You need four rotors of 1.2 m in diameter, with a solidity ratio of 0.1. With the $C_T$ of 0.008, we find a corresponding $C_P$ of 0.0007, which results in

$$P = C_P \cdot \rho A (\Omega R)^3 = 0.0007 \cdot 1.225 \cdot 1.05 \cdot 270^3 = 17.7 kW$$