Can we draw conclusions about content of thoughts from neural firing patterns? Can the neural firing patterns of animals be associated with specific kinds of thoughts? (ie Thoughts related to food, mating, or neighborly aggression?)

I am curious about the speculations in yesterday's Neuroskpetic blog post, "What is My Cat Thinking" on Discover Magazine's site. The author ponders if measuring his cat's neural firing patterns upon confronting a neighbor's cat, and then observing if similar patterns appear at other times when the neighbor's cat is not present, can be used as evidence for his cat's "thinking" or memory of the other cat. 

The author concludes, preemptively, that such an experiment would not demonstrate anything about the content of thought: "We have no idea how neural firing relates to consciousness in humans, let alone animals."

However, I am quite sure I have seen various papers published that use fMRI and neural patterns as evidence of particular states of mind. There are also social science experiments in which participants' brains are electrically stimulated, resulting in particular actions (grabbing of a pencil, for example), and then asked to explain the motivations for this action. These experiments would suggest that it is already assumed that specific brain patterns correlate to specific thoughts and actions.

I am wondering to what extent biologists have already studied neural firing in relation to consciousness and the content of thoughts. More specifically, can we infer the recall of specific memories from a neural firing pattern?
This is a complicated and loaded question. As Neuroskeptic noted, our understanding of consciousness is very poor (in fact, we don't know how to define it most of the time). To see some of the best current definitions, take a look at:
What are current neuronal explanations and models of 'consciousness'?
We definitely can't infer arbitrary properties of consciousness from something like fMRI. In general, we don't have a clear mapping from brain to mind, and thus all studies are inherently correlational in nature. Since all the work is correlational, it is thus not usually mentioned and the popular press tends to misconstrue this as a for-sure causal connection. But does that mean we can't infer any thoughts? No!
We can &quot;read thoughts&quot; well enough to produce an brain-computer interface. Adapted from my earlier answer:
Erik Ramsey is locked-in syndrome patient and is incapable of movement apart from his eyes. However, he has control of his brain enough to be distinguished by electrodes implanted into his cerebral cortex (Guenther, 2009). Although the sensors were implanted in the motor-area responsible for speech (which you would expect typical inviduals to have control over, since they can speak), the understanding of how to decode this is not full. Thus, the patient had to train in order to adapt to the system:

Accuracy of the volunteer's vowel productions with the synthesizer improved quickly with practice, with a 25% improvement in average hit rate (from 45% to 70%) and 46% decrease in average endpoint error from the first to the last block of a three-vowel task.

In a much more dramatic experiment, Adrian Owen has used brain imaging techniques as means of communicating with vegetative-state patients. It is known that visualizing playing-tennis and walking around the house produce very distinct (to an fMRI) activation in the brain. So Owen used that capability to allow a vegetative-state patient to answer yes-or-no questions. He asked the patient to imagine playing tennis for yes, navigating the house for no (Monti et al., 2010). Hence, neuroscientists think they can distinguish the thought of walking around the house versus playing tennis.
Of course, it is hard to say if this can be continued arbitrarily far, in practice or even in theory. For more info, see this question:
Is performance reducible to brain activity in an unambiguous way?
References
Guenther, F.H., Brumberg, J.S., Wright, E.J., Nieto-Castanon, A., Tourville, J.A., Panko, M., Law, R., Siebert, S.A., Bartels, J.L., Andreasen, D.S., Ehirim, P., Mao, H., &amp; Kennedy, P.R. (2009). A wireless brain-machine interface for real-time speech synthesis. PLoS ONE, 4(12)
Monti MM, Vanhaudenhuyse A, Coleman MR, Boly M, Pickard JD, Tshibanda L, Owen AM, &amp; Laureys S. (2010). Willful modulation of brain activity in disorders of consciousness. New England Journal of Medicine, 362(7):579-89.