How can I disable the "popping" sound when adjusting the volume?

You would have to turn the Notification sounds to off, or delete the specific sound that is played (but this would affect other event sounds and is not advised).

Screenshot:

It's not really documented anywhere, but holding ALT while pressing whichever key is responsible for volume up/down/mute will change the volume setting without the pops.

These are all controlled by gnome-settings-daemon, and unfortunately while VOLUME_UP_KEY and VOLUME_DOWN_KEY are assignable, VOLUME_DOWN_QUIET_KEY and VOLUME_UP_QUIET_KEY are hardcoded to ALT+XF86AudioLowerVolume and ALT+XF86AudioRaiseVolume respectively, with no easy way to change them.

I've opened a question regarding the feasibility of making the QUIET keys assignable in a future release.

TL;DR: You can hold ALT while changing the volume to mute the pops, but you can't currently assign this action to the default keys

The following command appears to work:

gsettings set org.gnome.desktop.sound event-sounds false