Can a neuron "remember" being stimulated? Can it remember how long ago, or what pattern of stimulation it received (for example, three times in one millisecond and then two times in the following millisecond)? If so, can it use this information to figure out whether or not to fire?

What do you mean, "remember?"

Preserve some kind of representation of how it was affected by its neighboring neurons in the past (insofar as that representation is useful to its proper functioning).

Can a neuron "remember" being stimulated? Can it remember how long ago, or what pattern of stimulation it received (for example, three times in one millisecond and then two times in the following millisecond)? If so, can it use this information to figure out whether or not to fire?

Neurons can adapt their coupling to connected neurons as a result of being excited (or not) by those connections. I learned the term Synaptic Plasticity in this context. Of course, a neuron does not directly work like a desktop computer. But from experiment it is known [*] that artificial stimulation of a neuron can change synaptic growth or shrinking depending on the frequency of the stimulus. So in this sense a neuron can also store patterns.

 

 

[*] That's only what I recall from attenting a Neuroscience lecture a few years ago. Sadly, I cannot offer a reference to the experiment.

You may want to look into long-term potentiation as well as regulatory changes upon stimulation (e.g. channel and receptor synthesis).