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They do when they are at the end of their life. Dogs and cats crawl away to die.

They do not do what humans do (now).

I have heard that cats may become so depressed over the death of a companion animal that they completely stop eating. This would certainly result in death if not corrected.

I used to keep pet crickets. When some crickets lose one of their back legs, they go and put their head under water and drown. So I guess it happens sometimes at least with insects.

Not in the way humans do. But crossing the road without checking traffic isn't all that smart.....but they don't know they will be killed for doing it.

I saw a video of a snake eating itself.

Yes. My fish tried to

My fish tried to kill himself as well. :( he kept trying to get sucked up in the filter

On accident.