Weight = (mass) x (acceleration of gravity)
Acceleration of gravity = 9.81 m/s² on Earth, 1.62 m/s² on the Moon.
The feather's weight is . . .
On Earth: (0.0001 kg) x (9.81 m/s²) = 0.000981 Newton
On the Moon: (0.0001 kg) x (1.62 m/s²) = 0.000162 N
The presence or absence of atmosphere makes no difference. In fact, the numbers would be the same if the feather were sealed in a jar, or spinning wildly in a tornado, or hanging by a thread, or floating in a bowl of water or chicken soup. Weight is just the force of gravity between the feather and the Earth. It's not affected by what's around the feather, or what's happening to it.