Electronic – Programmable FGMOS’ simulation model using Virtuoso

That is a mosfet used to make a very small capacitor. This is often done so that cmos process variation affect the active transistor and the "capacitor" equally. It also takes up fewer metal layers. What they are using the capacitor for is beyond me.

I suggest that to evaluate the correctness of your circuit design and your component value calculations that it would be useful to plug your circuit into a simulation program such as LTSpice to check its performance. If you do this before you build anything you can gain confidence that the circuit will give the response that you want.

In an application like you are attempting I would strongly suggest that you use an N-Channel MOSFET instead of the Darlington NPN transistors to drive the motor. This will give a high impedance at the gate so it is easier to design and get the timing delay circuit to work rather than the current bleed from the R/C part of the circuit that happens with the Darlington component.

Motors are inherently inductive and as such you will need to place a flyback diode across the motor leads to keep high voltage spikes from destroying the transistor when the motor goes off.

You may also want to consider introducing a voltage comparator between the R/C timing component and the motor driver transistor to be able to better set the delay timing threshold.