Q: How to use circuit analysis on a circuit ideal diodes with ideal op-amps
In comments you said you want to consider an ideal op-amp model.

For your second diagram, that means the op-amp's inverting input is at +1 V. It isn't at 0 V as you indicated because there's no feedback from the op-amp output to drive it to 0 V.

The node marked "-1V or 1V?" is also at +1 V by using Ohm's law on R3.

The node marked "????" is at \$-\infty\$ V because the ideal op-amp has infinite gain. 

What this really means is that an ideal op-amp model is not useful for analyzing this circuit. You should consider a model that includes saturation behavior so that the op-amp output can not be driven outside the power supply rails.