Choosing an OpAmp for MEMS mic preamp stage
I believe your chart choices are correct: the OPA344 was chosen for its low quiescent current, making it a good choice for battery-powered applications.  The trade-off comes from your other desired areas of improvement.

The OPA344 has a gain-bandwidth product of only 1 MHz, which makes it a poor choice for high-frequency amplification in your circuit.  In your circuit, an ideal op-amp's closed-loop DC gain from the resistors would be 100K/1.5K or about 67, and from your gain-bandwidth product of 1 MHz, your op-amp's open-loop gain is 67 at 15KHz (often considered the high end for audio).  The closed-loop gain is obviously lower.  The addition of a 100 pf capacitor in the feedback loop further decreases the high-frequency gain but keeps the op-amp stable with the phase shift to be experienced at high frequency at this high gain.  So this circuit will be fine for lower frequency audio, but the high frequencies will be attenuated.

Increasing the op-amp's speed will allow you to keep the gain and have better high-end response, at a cost of op-amp power.  As in all designs, the trade-offs are what makes it fun.

One other caveat, when using op-amps in audio circuits, beware of claims of "rail-to-rail."  If you dig through the data sheet, you will almost always find that an op-amp will not be able to deliver the same current near the rail; this can cause distortion with higher-amplitude signals.

Good luck!