I've got it now, thanks. So the main point is a general use arduino based voltmeter MUST have no connections to the outer world (this implies an isolated power source). So if I plug probe wires into the board's ground and A0 and put the probes on any two points in some external circuit the only way for the current to flow between the both circuits are the wires so all of the current will surely flow through A0. And since input pins are like 100MOhms the current is very limited.
As you have found powering things via USB can problematic.
There's more to be aware of. The metal case of a typical desktop PC is connected to the ATX power supply ground and the mains earth cable. If you have a second desktop PC then it will share the same ground as the first, so you cannot do a floating measurement on one whist the Arduino is powered by the other.
To make things more complicated, some laptops have isolated mains power supplies and some have grounded mains power supplies. If you are powering your Arduino from a laptop with an isolated power supply (check for the square within a square logo on the power supply) then you can do floating measurements. If the power supply isn't isolated then it's shares the same ground as your desktop PC.
Last quick quiestion to wrap it up then. Back again to the 5-12 fan example. What happens if I connect arduino voltmeter probes in reverse polarity so arduino ground gets 12v and A0 gets 5v? Is it allowed at all to feed negative voltage to input pins?
Negative voltages will fry the Arduino. If in doubt put a diode before the input and add 0.6v to whatever reading you see.