Electronically, how to control this 5v arduino-style relay board from a PC serial port
If by Electronically you mean by Software, yes you can Programmatically Toggle a Specific Serial Line. In LabVIEW this would look as follows:
Toggle a Specific Serial Line in LabVIEW
In Python you can use the pySerial Library, where the code would look like this:
__init__(port=None, baudrate=9600, bytesize=EIGHTBITS, parity=PARITY_NONE, stopbits=STOPBITS_ONE, timeout=None, xonxoff=False, rtscts=False, write_timeout=None, dsrdtr=False, inter_byte_timeout=None),
where rtscts=False sets RTS to Low and drsdtr=False sets DTR to low.
And you are right, the Serial Port Voltages can reach from -25 to +25 Volts, but in most cases it will be around -10 to 10 Volts.