Understanding the Kraakdoos Opamp circuit
It's nominally a noninverting opamp configuration with unity gain, although with a 10M feedback resistor, the feedback is not going to be very effective because of the opamp input's bias current and capacitance.

So yes, the opamp is running essentially open-loop with a feedback path that introduces enough phase shift &mdash; along with the phase shifts introduced by touching the various pads &mdash; to allow it to oscillate at various frequencies.

The output stage is a push-pull configuration using a complementary Darlington configuration on the top and bottom. However, it isn't biased to eliminate crossover distortion, so it will have a "dead band" in its transfer curve. This probably serves as a gating function that prevents low-level "background" signals from getting through to the output.