@donavabdear I assume you mean amplifier damping factor? I want to be sure we mean the same thing. For an amplifier, the damping factor is a defined figure referenced to a value of 8 ohms. Unfortunately, there is no requirement to publish it over frequency and it can increase for some amplifiers.
From a speaker point of view, a passive speaker, when tested, at least by main stream companies, is tested with a high damping factor amplifier, and even then, for accuracy, the signal is measured at the speaker so eliminate the amplifier from the measurement for frequency response. A high damping factor amplifier would also be used for distortion measurement.
For almost all solid state amps, the damping factor of the amplifier is so much higher than the internal impedance of the passive speaker due to the series impedance of the crossover and speaker driver, the the speaker, as a system, is relatively unaffected, at a gross level, by the amplifier. With most tube amplifiers, the damping factor is low enough, output impedance is high enough to have a significant system level impact on performance.
From an individual driver standpoint in a passive speaker, the impedance the driver sees changes as the frequency changes due to the series/parallel impedance of the crossover and that also changes with the crossover topology.
I am sure there are speaker cables with enough resistance to impact damping factor, but any normal speaker cable is going to have a very small overall impact on damping due to the series impedance of the speaker being large with a passive speaker.
When you connect a speaker directly to a driver, you eliminate the variable of the series impedance of the crossover. However, if you are using purely voltage drive, i.e. the output signal is just an amplified in voltage version of the input, then you still have the limitation of voice coil impedance limiting what people perceive as control of the driver. This is the limitation of connecting an internal or external traditional amplifier to a driver and using a line level crossover whether analog of digital.
That leads back to my discussion of why an advanced active speaker cannot be equalled, at least in linear motion of the transducer, by a passive version or a direct connected amp traditional voltage amplifier with electronic crossover, no matter whether inside or outside and no matter how expensive the amp. You are still working with the fundamentally limited voltage drive. Modern dynamic transducers are made to be as linear w.r.t. to voltage as possible. That is where much development has gone into w.r.t. motor structures (magnetics, magnetic components, voice coils, etc.). If you remove the restriction of pure voltage drive, and use combinations of voltage and current drive, then you can achieve more linear motion w.r.t. frequency. This can provide greater linearity over a wider operating range, and greater immunity to power compression. Current drive provides some inherent feedback (good!) that voltage drive just does not have and that is before getting into more sophisticated methods not to mention feed forward in the digital domain.
Rolling back to your second question. High damping factor ensures your speaker frequency response matches what the manufacturer publishes if they publish it. With most SS amps, the speaker driver and crossover impedance is so dominant that damping factor is meaningless w.r.t. "damping" the driver motion. There is no guarantee that the lower impedance is even the best dampening for the most linear motion. Think of a vibrating rod. Affix one end tightly, and it can keep vibrating. Hold one end lightly and it can keep vibrating. Hold it by the end with a mechanically absorbing material and energy is dissipated in that material and it quickly stops vibrating.
