The air in an elevator does tend to move with the elevator, because it has relatively little inertia. However, thinking about the problem in these terms seems, to me, misleading. The simplest way to think about this is to consider the acceleration of the elevator as and addition to the normal acceleration due to gravity.
In this light, it would be as if the helicopter were momentarily heavier wen the elevator accelerated upwards, and momentarily lighter when it accelerates downwards. This would inevitably cause changes in the height of the helicopter above the floor of the elevator, but I expect that most real-world elevators would not accelerate fast enough nor long enough for the helicopter to be smashed to the floor.
Of course, toy helicopters are not all alike, so your mileage may vary!
The first factor in stability is pitch inertia. Even though it is just a sheet of paper, its moment of inertia around the pitch axis is quite high, so any pitch motion is slow.
Next is aerodynamic pitch damping: As the paper starts to pitch, local forces at the edges will produce a stabilizing moment.
That the paper will slowly pitch up is due to the location of the center of lift, which will act at a quarter of the chord. The precise location is slightly ahead of the quarter chord, the more so, the more slender the paper is (ratio of span to chord length). This creates a pitch-up moment, and since the paper itself is rather floppy, will bend the whole paper into a graceful curve. This again helps to stabilize the paper, because a negatively cambered wing is naturally stable.
Your paperclip weight is needed to shift the center of gravity forward to one quarter of the paper airplane's chord. This is where the resulting lift force will act, so by shifting the cg forward, you minimize pitch moments. You could as well fold the first half of the sheet of paper into a tight roll of paper - this will produce the same effect as the paperclip. Once the pitch angle increases, so does lift and angle of attack. Beyond an angle of attack of maybe 10° or 15°, airflow on the top will separate, which will shift the center of lift backwards. Also, lift stops to increase with further angle of attack increases. Now the lift creates a pitch-down moment, which helps to regain the attitude at which the flow is still partially attached. Especially when using stiffer cardboard, this stabilizing effect of flying with a partially stalled wing is easy to reach once the center of gravity is right.
In folded paper airplanes, you will notice that a slightly bent-up trailing edge will help to stabilize the paper airplane, whereas folding it down will make sure it goes into a dive quickly. By folding the trailing edge at least partially up, you will create a local area of lower lift (or actual downforce), which will see a proportionally stronger increase of lift when the whole paper airplane pitches up. This results in a pitch-down moment, and vice versa. This is all what is needed to give a paper airplane positive stability.
A simple sheet will be too floppy to benefit from the stabilizing effect of a bent-up trailing edge, though.
Best Answer
But it does, however, some additional lift at the trailing edge from the downward-deflected trim tab remains and prevents the full return to the earlier position. What the trim tab does to the elevator, the elevator in turn does to the whole aircraft: Changing its pitch attitude.
See it this way: Before, the elevator was flying at its force-free angle. Then the added trim tab deflection changed the local pressure distribution on the elevator (in the case of a downward deflection it adds lift), so the elevator assumes a new equilibrium position. This new position is found when the lift change on the elevator just compensates for the newly added lift by the tab. Of course it must go trailing-edge up now.
Trim tabs (picture source)