Vertical autorotation with passive rotor
There's some info on autorotation, including some references, in this answer. And also a pertinent question on Physics SE. Free body diagrams included.

In short: 


Autorotation only works in a satisfactory way if the vehicle has forward velocity: in a vertical descent, in optimal circumstances, autorotation works as well as a leaky parachute does. 
Getting the rotor to actually rotate from standstill is a hair raising kettle of fish, the first thing about autorotation that helicopter pilots learn is that they must maintain rotor rotation from when it was driven by an engine. During any time of rotor standstill the aircraft is simply falling, with no real means of aligning the rotor with the airflow.
Autogyro's start their flight by starting the rotor to rotate. Can be done on ground by driving fast and making use of the horizontal velocity, or by a flywheel mechanism attached to the propeller engine. Again, the first thing to do is to rotate the rotor.


The equations in OP are valid for fixed wing aeroplanes, and for the rotor blades. But since with rotor blades the velocity over the blade is a function of the blade radius, plus faster over the forward moving blade than over the retreating blade, they are not defined relative to airspeed V but to tip speed of the rotor blades $\Omega R$. Impulse theory using non dimensional coefficients, from Prouty Helicopter Performance, Stability and Control page 25:

$$ T = C_T \cdot \rho A (\Omega R)^2$$

with


T = rotor thrust
$\rho$ = air density
A = disk area
$\Omega R$ = tip speed


Note that Prouty uses the USA definition for $C_T$, which incorporates the factor ½ of the dynamic pressure $½ \rho V^2$ and is therefore twice as much as $C_T$ values used in Europe.

For the thrust to drive the rotor, analogous to aeroplane drag:

$$Q = C_Q \cdot \rho A (\Omega R)^2 R$$

And for the power:

$$P = C_P \cdot \rho A (\Omega R)^3$$


  Also will all kinds of foils work in this scenario? What is the important parameter in foils that will make it start spinning and descent with constant speed?


Airfoil shape is of much less importance than blade twist, solidity ratio and a whole lot more parameters, which would make the answer very broad. Best looked up in Prouty and Leishman.