Electronic – Is it possible that I fried an MCU pin by performing a continuity test

The continuity function is designed to give an audible indication of resistance that is less than some threshold value.

To be useful, it will be designed to respond much more quickly than the display so that a tech can quickly 'buzz out' wiring and such like without waiting for the reading to settle or even taking his or her eyes off the test probes. It's specifically designed so that the voltage does not turn diodes on, it won't respond to resistors above a certain value and so on, so that it (usually) responds to just a fairly solid electrical connection.

The values will vary somewhat by manufacturer, but here is an excerpt from the Fluke 177 DMM manual:

As you can see it has hysteresis and a pulse stretcher that allows brief breaks to be detected. This is done by circuitry that is mostly operating in parallel to the main ADC function. Some crummy cheap meters have a continuity beep that is dependent on waiting for an ADC result, but they are not very useful. Avoid!

The updates in the question have helped clarify some points, although some concerns have not been answered, and the angle of the photos (not from directly above) makes it impossible to rule out connections to incorrect pins on the breadboard. However I can point out 2 problems so far:

• Schematic shows decoupling capacitors, but they don't exist on the photo of the actual hardware.

  Even if some 100nF capacitors were fitted to the breadboard (as mentioned in the updated text, but not shown in the photo), that does not meet the requirements shown in the ST "Getting started with STM32F030xx and STM32F070xx series hardware development" document (see section 5.4).

  This could result in various problems, but is unlikely (in my experience) to cause permanent hardware damage as reported. Even if the cause of the hardware damage is corrected, you might still have problems (perhaps constant or perhaps intermittent) until the decoupling is also improved to meet the requirements.

• No voltage is being supplied to the V_DDA and V_SSA pins (pins 9 & 8 on the LQFP48 package) - confirmed by the schematic and the photo of the breadboard.

  This is critical and these missing connections could cause internal hardware damage to the MCU. The datasheet and "getting started" documents explain more. Here is one example quotation to make the point clear (datasheet page 42):

  All main power (V_DD, V_DDA) and ground (V_SS, V_SSA) pins must always be connected to the external power supply, in the permitted range.

  On all the STM32 MCUs I have worked with, one common factor is that the documentation states you must supply power to V_DDA and V_SSA even if you are not using the ADCs. This is because an "analog block" inside the MCU (powered by V_DDA and V_SSA) also contains the power on reset circuitry which is always used, even if the ADCs are not used!

  Therefore as a minimum, you must connect V_DDA to V_DD and connect V_SSA to V_SS.

I have run out of time to investigate whether there are more problems and some things (e.g. power supply output spikes/ripple) have not been ruled-out. However that missing power supply to V_DDA and V_SSA needs to be at the top of your list of fixes, for your next attempt.

Read the documents that I linked to see the official ST recommendations for a minimal STM32F0 system.

Update: I am concerned about the missing connections to the ST Link/V2 which you mention. Some JTAG adapters require connections to the target's power, for their internal buffer ICs. I don't know about the ST Link/V2, and I don't have time to research it now. If I was in your situation, I would check this point.