Electronic – Is this op amp circuit using both negative and positive feedback

Did you use the model of the MC1458 in your simulation, or a generic (ideal) opamp? In the latter case that may explain why it doesn't work. The MC1458 is not a rail-to-rail opamp, and can't work near the rails both for input and output. Especially with a low supply voltage the operating range of the opamp is very limited, and is situated around Vcc/2. The Schmitt-trigger thresholds and your input signal are way below that.

Still, the output signal looks rather strange. What's the scope's vertical resolution? Anyway, I'd try a rail-to-rail opamp first, that's something which has to be fixed anyway, and see what results that gives. BTW, you could also bias the inverting input to Vcc/2 via a resistor divider and couple the signal in via a capacitor, but you'll still have a problem with the output levels. If you use the divider, don't forget to recalculate the Schmitt-trigger resistors.

If you do that the amplifier will amplifiy its own output and the ouput will go high or low and stick there until a sufficiently low or high input is presented, which will make it go the other way (low or high).

In essence you will have created a schmitt trigger.

What the formulas don't show is that the solution for this amplifier configuration describes the metastable point,

Another limitation of those equations is that they don't represent the limited output voltage range of real amplifiers.