Confusion about value of output voltage
Your understanding is correct. If the non-inverting input has -2 volts on it then, due to op-amp negative feedback, the inverting input will also have -2 volts on it. Of course the 1N4148 has leakage and this will bleed a bit of current through to make the -2 volts at the inverting input. I suspect that a better design would have a 1 kohm (or 10 kohm etc.) load resistor from the cathode of D2 to 0 volts. If this were implemented then you would see 0 volts at the output (within a few microvolts for a perfect op-amp).

If the non-inverting input were positive then the inverting input would follow that positive signal. It looks like this circuit is intended to be a precision rectifier or part of some sample and hold circuit.