Questions about Panasonic 12-32mm f/3.5-5.6 ASPH. Mega O.I.S.

Yes, I have it, it works. If you take a closer look at the photo at digitec, you'll see the thread just outside the label, with the 37mm label. Otherwise look here: http://www.panasonic.com/de/consumer/foto-video/lumix-g-objektive/h-fs12032e.specs.html

Yes, this lens is suitable for the G81, G7 and also all other cameras that use Micro Four Thirds (m43) interchangeable lenses. With the G81 you still benefit from dual stabiliser: the lens has a built-in stabiliser plus the G81 has one. For the sake of completeness: this lens also looks good on Olympus M34 cameras!