Which animals can suffer?

I think the functional approach is ultimately correct, but that suffering is a much more complex way of being than any system having learnable negative feedback. A simple way to illustrate this is to notice that the amount of suffering changes drastically (in my own experience) across functionally very similar negative feedback loops. Walking along a path involves negative feedback (which I can learn from), but I don't feel like I'm suffering at all when I notice I'm deviating slightly (and even if it turns out I am, the amount of suffering is still much much lower than for standard suffering). In fact, it suspiciously seems like the amount of suffering is correlated with experiences which would likely have harmed my reproductive fitness 20,000 years ago. It even disconnects from the sensation of pain: e.g. I suffer much less from painful experiences which I believe are healing compared to ones which I believe are harmful, even if the raw sensation feels the same. Another strange thing about suffering is that it increases the more attention I pay to a constant pain signal. On the other hand, emotional pain doesn't seem (in my experience) to be as separated from suffering. Anyway, the point is that we need to vet definitions of what suffering is in just our own experiences before trying to use them on animals.

You left out the category of possible answers "Such-and-such type of computational process corresponds to suffering", and then octopuses and ML algorithms might or might not qualify depending on how exactly the octopus brain works, and what the exact ML algorithm is, and what exactly is that "such-and-such" criterion I mentioned. I definitely put far more weight on this category of answers than the two you suggested.

If an animal can learn to avoid pain (not just avoid it right now, but memorize what happened at given conditions and avoid these conditions), then the animal can suffer.

Is it any easier to determine if an animal is in "pain" than if it is "suffering"? The quotes are because I can't be sure what you mean by these words, which have a range of uses. "Pain" is sometimes broadened to mean the same as "suffering", and "suffering" broadened so far as to mean "not getting what one wanted."

How do other entities fit into this, such as plants, self-driving cars, and configurations in Conway's Life?

Presumably people think that at some point an AI is able to suffer. So why wouldn't a neural network be able to suffer?

You have to define "suffer" more precisely.  Which is the whole problem with qualia and experiential judgement.  Personally, my answer is "anything with two neurons rubbing together can feel reward/loss results".  And I'm not sure other humans feel the same things the same way I do, let alone flatworms (though I generally stipulate that humans are close enough to reason about). 

It seems likely (based on priors and analogy, not any measure I can think of) that, as brains get more complicated, that emotions and experiences are more intense.  If so, one could set an arbitrary threshold of complexity for what one labels "suffering".  But why?  The question should be "what and how much does X experience, and how do I aggregate that experience across beings in my moral calculations?"  I don't have an answer, and I don't currently believe that there is an objective answer.