What is the best way to monitor audio on the GH2, given that there is no headphone jack? All I could think of is to use an external video monitor that has headphone jack.

You don't. It doesn't output audio via HDMI.

Watch levels and play back takes where the audio matters to verify

What about outputting via the AV output? Or does that not work live?

It's possible if it works during recording but be aware that the AV out audio is line level not headphones level. You will need an adapter to lower the impedance otherwise the sound will be horribly distorted.