How are individual neuron firing rates characterized?
However, upon reviewing these papers, it appears that the neurons were
  stimulated while still inside the tissue, but after the tissue had
  been sliced. However, this seems so absurd that I must be misreading
  it.


You're not misreading it. I can't access the paper you refer to, but this is a standard way to measure the electrophysiological properties of individual neurons, and has been for more than 25 years. (I'm really curious what you find "so absurd" about this).

I'll take an educated guess (since I can't read the methods online) that these recordings are from rats (or possibly mice), using whole cell patch clamp technique in thin slices of the young rat neocortex. Briefly put: the animal is sacrificed, the brain is removed, it is sliced into about 400 micron thin slices using a vibrating razor blade, and the slices are kept in a artificial cerebrospinal solution that allows the neurons to remain functional for a day's worth of experiments. Then individual neurons are imaged under a microscope and "patched onto" with a glass micropipette that contains a silver wire. Electrical events in the neuron can thereby be recorded into one's computer and plotted as shown in the figure you provided.

In this case, what's shown are what are known as current clamp recordings, in which current is injected into a neuron and the resulting voltage changes are recorded. Here, they are injecting a depolarizing current to make the neurons spike (have action potentials) and seeing that the neurons they sampled vary in their spiking properties. Some are fast spiking, some are slower spiking, some have different spike waveforms or patterns of spiking.