STM32F7 HSE not starting
My first question, before any further detail, is: how can I find out
if the crystal is actually oscillating properly and is not broken?

Get an oscilloscope and put the ground of the probe on either pins 2 or 4 of the oscillator (make sure you make the ground of the probe as short as possible). Then put the oscilloscope probe on pin 3 (while it's grounded). You should see a nice clean square wave with the appropriate voltage levels described in the datasheet.
As far as as cube mx goes make sure you also select the appropriate clock source in the GPIO configuration.