What is the physical scale of the turbulence that affects large passenger planes?
Summary

There are in effect no minimum levels of displacement that will not have noticeable effects. Even very small displacements, of an appropriate period and spectrum, can affect a large aircraft. Scale alone is not a limiting factor.

In more detail

Question 2: Are there any minimum limits, so that even fiercely turbulent air below a certain displacement is not noticeable?

The title addresses "large passenger planes" and not passengers, or passengers with vestibular anomalies, etc.

Generally mountain wave is considered "fiercely turbulent" as the updrafts and downdrafts can exceed the climb and descent capability of an aircraft.  However, the "frequency component" of the mountain wave can be very low, at times creating no human perception of motion. Yet the turbulence causes loss of control by virtue of exceeding the aircraft's capabilities of counteracting an upsetting force.  

So in this instance, it is possible to have an extreme situation, and actually loss of control, by turbulence which is imperceptible by a human without instrumentation.

Question #1: What magnitude of displacement of air in turbulence conditions is large enough to affect (i.e. be noticeable in) an airliner?

The aircraft, and it's contents (including people) act as a system. The system can dampen the effects of some forces and cause amplification of other forces.  

A tiny, repeated push to a child on a swing, can result in very large displacements of the swing, and is a good example of a form of resonance or amplification of the effects of a small push.  The pushes have to have the right periodicity, matching the resonance of the swing to be effective. 

The above is an example of a resonant system, but we also have around us systems which dampen energy, such as the suspension on a car.  Driving a car over a bump in the road might create a very mild sensation in the car, but if we were riding on a solid wheel hitting the same bump at the same speed the resulting bump could be injurious. 

So to only ask about displacement is akin to only asking about sound pressure of music. The frequency (notes) help create the music, and the periodic occurrence of the frequency energy also helps create the music (like beat). A drum strike can be very loud or a triangle very soft, but they are both noticeable in music, in part due to our perceptual training, but also due to the human auditory system.

So for an aircraft, a displacement at one frequency may have essentially no impact on the flight path, due to the systemic response of the aircraft. Furthermore that displacement may or may not be detectable by a human riding in that aircraft.
 To answer your broad, but limited to displacement question, becomes impossible to do so accurately if you choose to limit the dimensionality of the system you are seeking analysis of. (reference your comment)