Is this input-output relation correct? Schmitt trigger
OK, operational amplifier will want to keep node A at GND level. It
  will try to set its output so that this(inverting input potential =
  non-inverting input potential) happens all the time.


It's not negative feedback - it's positive feedback. That's how a schmitt trigger works.


  So, why isn't a 5V of input mapping to -10 V which will hold node A at
  0 V


It's not negative feedback. It's a non-inverting comparator with hysteresis.