Questions Regarding Metal Detector Tx Coil Circuit
The frequency of the tank is about 27KHz, which I'm guessing is the
  fundamental frequency of the astable.


No, the op-amp isn't acting as an astable multivibrator; the 100 nF capacitor fed from R5 (100 kohm) is there to self bias the op-amp so that it produces a pretty decent 50:50 square wave. The 100 kohm and 100 nF provide a cut-off at about 16 Hz so are way, way below the operating frequency of the tank.

The transistor output stage is inverting (collectors at the output point feeding the tank) and this means the sinewave signal from the tank gets turned into a square wave by the op-amp and the transistors force the tank in the opposite direction. This causes sustained oscillation.

Your feeling about the way the output transistors are driven appears correct - it's just a way of saving a few mA and extending battery life.