The analogue input pin impedance for the STM32F030 is listed as around 50K in the datasheet. Does this value apply only during the conversion period, or would it be higher when the pin is configured as an analogue input pin but no conversion is taking place?
You've misread the datasheet. RAIN is the maximum external impedance that can be connected to the ADC pin in order to keep error below 1/4 LSb on a calibrated ADC.
The impedance of the S+H circuit is RADC, listed below that. The impedance of the circuitry beyond the S+H circuit is not given, but based on a RAIN of 50kohm with an error of 1/4LSb with a 12-bit ADC, I do not expect it to be any less than approximately 820Mohm.
See figure 23 for details.
Based on AN2834 - https://www.st.com/content/ccc/resource/technical/document/application_note/group0/3f/4c/a4/82/bd/63/4e/92/CD00211314/files/CD00211314.pdf/jcr:content/translations/en.CD00211314.pdf - How to get the best ADC accuracy in STM32 microcontrollers - Figure 15 - what the OP is looking for is Radc which is 1K. From the different equations given, you can't see beyond the switch impedance.
The app note goes into lots of detail on how they system works and how to best interface to it.