Does comparing neurons firing with beliefs/desires involve a type distinction that renders belief/desires irreducible to patterns of neuron activity?
I don't think it does, but I do think that the difference in scale between a neuron firing and an entire belief forming makes the reduction very difficult, and possibly pointless. It's a bit like reducing the spray of water out of a garden hose to the movement of individual water molecules. It's very difficult, given that each water molecule's motion contributes only an infinitesimal amount to the movement of the water as whole. Furthermore, even if you could map particular interactions between water molecules to specific motions of water droplets or the water stream as a whole, would you learn anything new thereby? Would it help solve any of the problems you're interested in? A lot of the time, it's better to work at higher levels of abstraction.
How folk psychology relates to the scientific picture of mind and world, was a perennial question of philosophy of mind, long before it was dubbed Bermúdez's "Interface Problem" (a name for it that I never heard before). Eliminativism (Dennett: folk psychology has nothing to do with reality), various identity theories (e.g. Armstrong: the mental states of folk psychology are all brain states with particular causal roles), the peculiar option of "anomalous monism" (Davidson: mental properties are real but have no systematic relationship to physical properties) are among the schools of thought.
As far as I can tell, Less Wrong doesn't have an official answer to this question. There has been much more perplexity here about qualia, than about intentionality. Regarding intentionality, the implicit school of thought here might be functionalism with compositionality, i.e. brain states are combinations of semantic parts that get their meanings from their causal roles; with the details of this being a contingent computational technicality. This is a philosophy made to apply to computer programs as well as to living brains.
Personally I have argued, on the grounds of realism regarding consciousness, that ordinary computational theories of mind are not viable for conscious minds, because, from a physical perspective, neural computations are based on virtual state machines that are ontologically a little fuzzy, and the individual consciousness is objectively distinct. So I have posited that the self corresponds to something like a large entangled tensor factor in the ultimate quantum-gravitational ontology of the brain, but with the understanding that this mathematical description is a little hollow, and that its true ontology would be more like something out of transcendental phenomenology...
A long time ago I argued this here, calling it a non-naturalistic quantum monadology, but didn't get many takers. I mention it now because you did ask, and I expect that the truth must be more like that, than like the simpler answers you'll get from existing theories.
I hope this question makes sense although given my confusion it has come out as word vomit and not a pointed question. So sorry for that.
How does commonsense psychological explanation (folk psychology) interface with the explanations of cognition and mental operations given by scientific psychology, cognitive science, cognitive neuroscience and the other levels in the explanatory hierarchy?
Does comparing neurons firing with beliefs/desires involve a type distinction that renders belief/desires irreducible to patterns of neuron activity? Is one a belief and the other a belief about a belief? Which is the map and which is the terriotory, or are both the map or both the territory? I am entirely lost.