Yes, the A7III is an E Mount lens mount. 

Yes it should work with the sony a6000. As long as the ka lens has an aperture ring on the lens and is not solely dependent on receiving aperture control from the body, this adapter should work fine with ka lenses. To my knowledge, there are no sony e mount to k mount adapters with electronic coupling so no lens control through the body (meaning aperture and focus are fully manual). 

This 100% works on any of the A7 series. it works on my A6000, a7r2 and a7r3. If the body is E-mount it will work. If the body is A-mount it will not.