Should I connect unused STM32 pins to ground?
No.

Usually, the data sheet explicitly mentions what to do with unused pins. For most microcontrollers, the correct approach is to leave them open on the PCB, and leave the internal pull-up logic enabled after boot (the initial state is safe, so leaving it is safe, too).

It is generally a bad idea to connect I/O pins to a power rail without an in-line resistor. If you ever enable one of these pins as an output, the pin will drive against the power rail, and lose.