It's illogical that I think it is unnecessary in North America for the poorest of the poor to feel the need to eat their dogs?
Yes. Grotesque as we may see it, if people are desperate enough to sell their bodies for abuse sexually, or their children, why would we stop for dogs?
There are quite a few problems with your analogy. Most importantly, it is completely off-topic because my argument in the above quote is that even the poorest people in the US have sources they can go to for free food, and so eating a dog is unnecessary. I don't think this would be hard to prove.
Secondly, there is an important distinction between self-harm and harm to others.
Thirdly, you are assuming as a premise that it is acceptable for people to sell their bodies for sexual abuse. I have not expressed any opinion on this either way.
Or it is illogical that I think people who do so unnecessarily are mentally sick?
It is an opinion, but to make such an assertion as the basis for an argument or in the context of an argument is illogical. For starters, it introduces an ad hominem attack - you aren't attacking the notion of eating dogs in and of itself as much as the people who do so.
It would be an ad hominem fallacy if I were to write, "person A ate their dog, and person A is mentally sick, so what person A did must be wrong." (I feel sick almost writing about this...)
Also, I am attacking both the people who partake in something like that, and the action itself. This is not illogical.
Or it is illogical that those who sympathise with unnecessary and disturbing actions are likely to be sick themselves?
Not if the foundation of your argument -- that the actions are unnecessary and disturbing -- is sound. But I don't believe you've shown it to be so.
First, it should be obvious to most people that the actions are unnecessary - we have food banks, etc. And you really need further motivation for the disturbing part? That is what bothers me so much.