Let's say, for example, if a helicopter that has an unlimited fuel supply and everything else needed to stay functional stays afloat in air for long enough, would it eventually start moving backwards? When an object comes off of the surface of the earth, it gets the earths 'velocity' (I didn't know what else to call it, it's been a while since I've been in physics and I don't remember much about rotational kinematics), but soon enough wouldnt the helicopter start losing that speed and start slowing down/going backwards due to friction from the air? Or would the air be kind of like pushing on it to keep it from doing that?
The atmosphere moves along with the earth's surface at the exact same speed. This means the helicopter will continue to stay in the same position above the ground. There is no reason why it would "move backwards", unless there is wind that would push it backwards.
When an object comes off of the surface of the earth, it gets the earths 'velocity'
This is because of conservation of momentum. If an object is on the surface of the earth (or any uniformly moving body like a train moving with constant velocity, for example), and then takes off vertically, it will have a component of velocity parallel to the ground, equal to the speed of the ground.
but soon enough wouldnt the helicopter start losing that speed and start slowing down/going backwards due to friction from the air?
If the helicopter is moving horizontally, provided it maintains the same thrust, it should keep the same the same speed, and will not move backwards.