I don't think there's a great way to test all of the functionality of the MCU at once. The parts that I'd first suspect would be the I/O drivers and the crystal. I'd suggest programming it to output something like a square wave on each pin, and then use a second microcontroller to verify that the voltages are being output. Double-check that the supply current is within reason, as often the ESD diodes are the first thing to go, and even though the IC could continue to work, it'll generally use more current overall.
Finally, use the tester to measure the output frequency of the STM32 while the STM32 is in the mode where it's programmed to use the external clock crystal. Make sure that it's within 75 ppm (or so).
In other words, it's probably too much trouble to "check if they are fully functional", so you probably should just try using them. If they work at all, they're likely fine.
I usually don't use Arduino, but their libraries are generally set up to be very easy to use. But, Arduino does impose some performance limitations. (I've been using SystemWorkbench for STM32, an Eclipse-based IDE. It's fairly complicated but it does include a debugger. My understanding is that the Arduino IDE doesn't come with a debugger.)
As already mentioned, the boards usually don't come with a bootloader, so you'd need to use either an ST/Link or a TTL-level serial port to program them. You may need to set the BOOT0 jumper in order to run the ROM bootloader. Note that with BOOT0 set, your programs may mostly work, but may fail since the bootloader will overwrite bits of your code. Be sure to un-set the BOOT0 jumper before running your software.