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An octopus? A crow?

These are two of the smartest animals in the animal kingdom
Taxonomic classifications are built out around recognizable similarities in animals biological structures. Setting aside how precisely we understand brain function in non-human species, it takes little effort to see that most vertebrates are structured in a way that heavily overlaps with human biology. They are similar. Why would you assume they aren't?
And if you go from a "philosophical" perspective and ask how we could possibly know what they feel, you may well extend that to humans. How do I know you experience qualia?
You're getting so caught up in precision that you're overlooking obvious similarities. Can't see the forest through the trees. That animal has a nervous system and demonstrates a pain response. How much more detail do you need to be able to understand it feels pain?
For creatures without similar structures, we don't make the same assumptions about pain and suffering. Do carrots feel fear? Do mollusks weep? No one thinks about that. >

Except for the fact that many people claim to empathize with invertebrates or even plants. You clearly understand enough biology to think clearly about this topic. All I'm saying is that everyone else should make an attempt to meet that minimum standard of understanding. >


many people claim to empathize with invertebrates or even plants.

Very few claim to empathize with invertebrates, but crabs and lobsters and other crustaceans have pain responses too, so it's still not an example on your side here. My point was that people see similarity in responses and behavior, and that's made possible by our very real similarities. That Doctor so-and-so knows the science of it all and Dennis your vegan high school classmate does not makes no difference. 
The science explains why we have similarities and why we see and feel similarities that allow us to empathize. But not knowing the science doesn't invalidate the emotional feeling—the right answer is right whether or not you show your work.
As for people who "empathize" with plants, they fall into two categories: [1] People who are personifying plants because it's a useful rhetorical technique, but don't actually believe plants feel complex emotions; and [2] extreme outliers that don't merit serious consideration. If "crazy people believe crazy things" is your best defense here, you've lost focus on the "philosophical" question your post is about. >

We might just have different experiences when it comes to people's claims about the range of their empathy. I know plenty of not-crazy individuals who either claim to or speak as though they understand the feelings of an invertebrate. I might have a sampling bias because I spend a lot of time around animals and teach HS biology. 
I agree that the right answer is right whether or not you show your work, but showing your work is necessary to distinguish between the right answer and an answer that seems right, but isn't. It certainly seems like it'd be painful to be a clam being eaten by an octopus. This might even illicit a little bit of sadness for the clam... But upon further reflection I might realize that clams don't have centralized nervous systems. They don't have families that will mourn their demise. They don't have unfulfilled dreams or projects. I've now shown my work and realized that my previous answer was incorrect. >

Again, you're pushing out to extremes to justify your position, but if you have to run all the way to mollusks, animals without a central nervous system, to stick to your view, it sure seems like you've abandoned it for every animal that does have one.
And for those, there's no need for people to show their work. Because they're right. Someone else can check it for them. Verified. Instinct to empathy validated. >

Not necessarily. I used mollusks because it's a simple and easy case. I can make my point without getting lost in the details. We can ask the same question for animals of increasing complexity. You begin at simple cases, then move on to more difficult cases. >

When an animal or insect is trying to escape the stimulus, that's an indication that it's at best, 'bad' and at worst 'very painful.' It's quite easy to see when what you're doing causes an animal pain.
And as far as having empathy for a spider: maybe having a leg ripped off hurts, maybe it doesn't (see my last point below), but either way it's no longer at full-capacity, which is a situation that a person can empathize with: it's not unreasonable.
You say not to over-estimate pain receptors, but if the pain level is unknown, what about under-estimating them?
You also say it's complex and not-understood, but then you say to follow your understanding of it. If it's not understood, why is your understanding correct? >

Why does an attempted escape necessarily indicate pain? Even microscopic organisms move away from stimuli that aren't conducive to their existence. 
You can't empathize with a spider being at half-capacity because the human experience of a handicap is fundamentally different from the arachnid experience (if arachnids experience anything at all). At best you'd imagine a life where it's slightly more difficult to acquire food (minus the anxiety and hunger pains and social stigma and the plethora of other uniquely human experiences that cloud our judgement when empathy enters the picture). >

You say not to over-estimate pain receptors and consciousness, but if pain levels and the nature of consciousness is unknown, what about under-estimating them? And why do you think your correct about things that you say are unknown?

At best you'd imagine a life where it's slightly more difficult to acquire food

That's what I mean; at minimum, there's this to empathize with; it's not nothing >

I don't think that consciousness or pain is "unknown". You're conscious. You feel pain. You can empirically study these phenomena. We're far more likely to overestimate pain levels in non-human animals because humans are subject to greater levels of pain in a wider variety of "flavors." Underestimating the pain of non-human animals is certainly a danger (ask Descartes), but I think it's a less common one because of how complex our nervous system is. 
To be clear: I said that you ought to drop your pretense of knowledge, not that the topics are intrinsically mysterious. It's helpful to begin with a blank slate when studying complex topics because outside assumptions can make proper learning difficult. 
But that experience, the experience of having trouble finding food, is intrinsically linked to our humanity. Spiders don't have mammalian stomachs that rumble. They don't have the same receptors that send hunger pangs. They don't have salivary glands that cause their mouth to water when they think of food. They don't have hands that get shaky when they're malnourished. When you strip the experience of all the anthropological features, you're left with nothing. >


humans are subject to greater levels of pain in a wider variety of "flavors."

Can you explain this, please? I think I disagree, but I want to know what you mean first.

Underestimating the pain of non-human animals is certainly a danger (ask Descartes), but I think it's a less common one because of how complex our nervous system is.

Therefore, empathy is not a poor tool when thinking about animal suffering?

I said that you ought to drop your pretense of knowledge, not that the topics are intrinsically mysterious. 

If there aren't concrete answers and measures, there is mystery. If this wasn't somewhat vague / mysterious, we wouldn't need this CMV

When you strip the experience of all the anthropological features, you're left with nothing.

You talk about the complexity of nervous systems, but then have this very reductionist conclusion here about being 'left with nothing'? >

Humans experience greater levels of pain when responding to similar stimuli. Our sensory organs (especially our pain receptors) are very finely tuned compared to most other species (obviously there are exceptions like the eyes of hawks, the noses of dogs, and the ears of bats, but generally this holds true). In addition to this, humans experience social and psychological pains. We experience rejection, depression, anxiety, shame, and a whole slew of other negative emotions. Other animals (especially social ones) experience some of these emotions, but humans do something uniquely masochistic: we dwell on these pains. We have minds that can become torturous as they replay the most painful moments of our lives. 
I don't think that empathy has anything to do with general estimates of the capacity for pain of non-human animals. 
Concrete metrics aren't necessary for a mystery to be dissolved. You can have clarity about a topic without rigorous measurements. This is why we have things like literature, poetry, music, and philosophy. 
Yes. Exactly. The complexity of the nervous system is precisely what gives rise to the layers of an experience. If I move from one nervous system to another totally different one, I lose those layers. >


Humans experience greater levels of pain when responding to similar stimuli

This sounds… questionable. What’s your source for this?
From my admittedly limited understanding of psychology, this doesn’t even sound measurable. You can only measure pain via response to pain (i.e. behaviour), or via cognitive imagery like EEGs.
Neither of these allow for comparison with animals. Animals can exhibit very different behaviours from humans despite the same stimulus (more basic example: dogs pant instead of sweating, doesn’t mean they feel less heat), and brain imagery is obviously only applicable to humans.