The CP2102 was connected to the STM32 GPIO pins while USART1 was not configured (STM32 clean / not programmed). Is it possible that I destroyed my CP2102N via the GPIOs of the STM32 in that "initial state"? Would that be enough to damage the CP2102? Or why else doesn't my PC recognize the CP2102 at all?
You have not correctly wired the CP2102 for USB operation.
Among multiple possible issues:
the required connection of the VBus input is missing, which prevents the chip from placing the virtual enumeration resistor. To quote the data sheet "VBUS Sense Input. This pin should be connected to the VBUS signal of a USB network. A 5 V signal on this pin indicates a USB network connection."
you have feed REGIN with 3v3, but for a CP2102 the minimum to use that is 4 volts (the CP2109 apparently can accept down to 3 volts there)
the resistor you have on VBus of the USB connector likely accomplishes nothing. Hosts do not detect peripherals based on power consumption, but by the presence of a pullup resistor on a data line
Overall, you should review your design in comparison to manufacturer examples. Anything you do differently will need to be based on a very strong understanding of the chip, and you should consider if your belief that you could do that is actually mistaken.
As this is a QFN chip, also carefully inspect soldering under 10x magnification from an angle, so that you can see each of the solder fillets between the chip and the board. An eye loupe can work, or raise your microscope up and tilt the board at various angles in your hand. Also check trivial things such as that the orientation of the chip is correct, and that the pinout didn't get mixed up in the schematic design.
When the CP2102 part of the circuit is correct, you should get enumeration and ultimately (with correct drivers) a serial device, irrespective of just about anything but a power-to-ground short on the STM32 side of the board.