Is there any chance that a neuron could fire when hyperpolarized? In that case, would the spike be different than usual?
Perhaps this is not what you asked, but there's a phenomenon called rebound spiking or postinhibitory spiking where a hyperpolarization causes spiking. This is due to the oscillatory property of membrane dynamics (certain subsets of type-II neurons). Spikes can be evoked after inhibitory current stops. Figure 7.29 from Izhikevich's book:
Here's a partial figure from Hasselmo 2014 (recordings from entorhinal cortex stellate cells).
Hasselmo, M. E. (2014). Neuronal rebound spiking, resonance frequency and theta cycle skipping may contribute to grid cell firing in medial entorhinal cortex. Phil. Trans. R. Soc. B, 369(1635):20120523+.
Izhikevich, E. M. (2007). Dynamical systems in neuroscience : the geometry of excitability and bursting. Computational neuroscience. MIT Press.
In principle, no. An action potential is initiated due to the activation of Na+ channels. These are so-called voltage-gated channels, meaning that they sense a depolarization and subsequently open their Na+ pore to allow Na+ to flow in, further depolarizing the cell. This is the depolarizing phase of an action potential.
Voltage-gated K+ channels open up also due to depolarization, but have slower kinetics and hence they kick in later than the Na+ channels. Influx of K+ opposes the effect of Na+ influx and brings the cell membrane potential back to its resting state (or even hyperpolarizes the cell). This ends the action potential (Fig. 1).
Fig. 1. Action potential and membrane conductances. source: Antranik
An action potential cannot be generated in a hyperpolarized state, because the Na+ channels are closed. A depolarization is necessary to open them.
If a neuron was able to fire when hyperpolarized, there would be no regulatory mechanism to inhibit the cell. This would mean it would keep on firing, regardless of its input - not very useful.
So far, we're still missing some interesting cases in your answers. Izhikevich has developed a strikingly simple model that covers the following, including several inhibition-induced action potentials, and there are specific cells that have been discovered matching these behaviors:
M) Rebound spike (as has been mentioned)
N) Rebound bursting
O) Threshold variability
S) Inhibition-induced spiking
T) Inhibition-induced bursting
K) Resonators, if I recall, can be frequency responsive to inhibitory signals
P) Bistable, again, if I recall, bistable neurons can be switched with inhibitory signals
His textbook Dynamical Systems in Neuroscience (and I'm sure several papers, or references) goes through the names of specific cells, for these, such as: Stellate Cells of Entorhinal Cortex exhibit rebound spikes and Spiny Projection Neurons of Neostriatum exhibit bistability.