I'm assuming all but the final two situations are with the effect epicenter on the surface of the Earth in an open area.
1 cubic foot, 10 cubic feet, 100 cubic feet
Gravity is gone inside a relatively small area. The air pressure inside the volume might be a little wonky, causing a perpetual updraft. The air at the edges of the volume, especially the bottom, would be under more pressure than the air just inside and it would pour in and flow upward to the slightly lower pressure air on top.
1 cubic mile
The ground half a mile down is under a fair amount of pressure from the ground above it (about 200 atmospheres). As the device turns on the pressure is immediately relieved and the ground will pop up rather violently. There is no gravity to stop the upward motion and the ground will reach the top of the volume and then oscillate at the edge. Moving past the edge and falling down to collide with ground moving upward. Eventually any floating ground material will pile up at the edges outside the volume of effect. This would be a great way to do some strip mining.
10 cubic miles
At 1.3 miles down the ground has 645 atmospheres of pressure from the ground above it. When the device is enable this pressure is immediately released and the ground violently explodes upward, likely killing anyone on the surface. The ground will rapidly reach the upper edge of the volume and be flung in all directions. Air rushes into the sides of the volume and blows out the top into the low pressure atmosphere above. The heavier than air materials will eventually settle around the edges of the volume.
100 cubic miles
Similarly to the 10 cubic mile case, except the pressure released is over 1,400 atmospheres at 2.98 miles down. The explosion is more violent and spreads material even further away. The volume cycles huge amounts of air through it and significantly disrupts the local weather.
encompassing the entire surface of the earth and atmosphere, but not
penetrating into the surface
We all die. We lose significant air pressure and people would asphyxiate immediately. Everything not bolted down is now airborne as it's no longer tethered to the Earth by gravity it continues along its path. You would observe this as the the Earth and everything bolted down as pulling away from you and everything not bolted down. Everything gets a turn at exiting where the edge of the atmosphere used to be then being pulled back down to collide with the surface of the Earth again. Masses of vehicles, boats, and bodies will begin to form into huge hurricane spirals due to the coriolis effect. The floating debris begins to move at different speeds than the surface of the Earth as it spins. Eventually everything gets dumped into an ocean (assuming the effect hugs the surface of that too, otherwise it's flying around with our bodies).
encompassing the entire earth, from the top of the atmosphere right
down to the core.
The Earth explodes. The core is under immense pressure from the weight of the ground above it. Suddenly that weight is gone and that pressure is going to equalize. The mass of the Earth ejects from the edge of the anti-gravity volume but there is no Earth to maintain the gravity outside of it. It continues outward but is eventually pulled into an oscillating dance at the edges of the sphere while pieces on the far side pull on pieces on the opposite side. As they fall into the anti-gravity sphere they continue through where the Earth once was at constant velocity (unless they hit other pieces doing the same) and emerge to be pulled somewhere else. Oh, and obviously, everyone is dead.