Phase-locking can occur at very high frequencies at the early stages of a sensory system. In addition to the answer provided in your previous question, let me offer an illustrative example from single neurons in the somatosensory system.
Deschênes, Timofeeva and Lavallée (2003), have stimulated the whiskers of an anesthetized rat with a sinusoidal stimulus while recording extracellularly in the main feed-forward brainstem nucleus of this system (Pr5). Practically all neurons responded with at least one action potential to all cycles of a stimulus up to a stimulation frequency of $300Hz$. Neurons always fired an action potential at first cycle up to the maximum frequency tested, $400Hz$, and ~$30\%$ fewer spikes upon subsequent stimulation (questions 2 and 3). The extend of coupling to the stimulation frequency depended on the amplitude, velocity, and direction of the stimulation.
Encoding of high-frequency whisker deflections by primary afferent axons
(Figure 6 from Deschênes, Timofeeva and Lavallée (2003))
Given the physical limitations of the stimulation device (piezoelectric element) it was not possible to directly stimulate at higher frequencies. However, the authors took advantage of the harmonic oscillations of the stimulator to look at the phase-coupling of spiking to higher frequencies of stimulations. Remarkably, units that were sensitive to small displacements of the whisker were able to follow the harmonic oscillations of the stimulator up to at least $1kHz$ (see Figure above).
What happens beyond? Does the neuron follow a virtual stimulus of 1/n of the original frequency (which would be in phase with the original stimulus)? Or does chaotic firing take over?
It is important to note that $1kHz$ is very close to the refractory period of most neurons, thus I would predict that this could be the only limiting factor in these cells. Of course, different types of cells will show different responses to high-frequency stimulation based on their intrinsic properties, the degree of short-term synaptic plasticity of the activated synapses, and/or the filtering of the incoming inputs by upstream circuits.
Does this maximal frequency count as a characteristic of a neuron (type of neuron), and what is its name?
I am not aware of something like this. Stimulation at such high frequencies is not typically used to classify neurons as it is rather non-physiological for neurons beyond those receiving direct inputs from the sensory receptors. Most often researchers apply a DC pulse and observe the degree of spiking adaptation.
Deschênes, M., Timofeeva, E., & Lavallée, P. (2003). The relay of high-frequency sensory signals in the Whisker-to-barreloid pathway. The Journal of Neuroscience : The Official Journal of the Society for Neuroscience, 23(17), 6778–87. https://doi.org/23/17/6778