Since it is safe to say that animals’ capacities for language and abstraction are at the lower end of the spectrum, it is also safe to conclude that their level of self-awareness regarding thoughts or instincts will be fairly low.
The language thing is crucial. In short, no animals have it. Not even chimps - they can learn a lot of vocabulary, but they never get grammar. A chimp apparently doesn't get the difference between "I sit on chair" and "Chair sit on I." This hugely limits animals' capacity for abstract thought.
Another question: are any animals aware of their own mortality? My guess is they are not.
Just how little thinking capacity they have I saw illustrated years ago when I read about a case where a barn caught fire. There were horses inside, so the farmer opened the doors for the animals to get out. They refused. He eventually managed to chase out most of them. And then, incredibly, they ran straight back into the inferno and many got fried.
Why? Because horses don't think at all. They did not realize it was the fire that caused their pain. All they knew was that there was big trouble, and the barn was the one place where they always felt safe.
Another probably germane point to bear in mind (pun intended) is that all neural processes involving consciousness appear to be threshold dependent—that is, there’s some minimal level of neural activity of a given type must be exceeded before the attendant awareness is kicked awake. Pain is an archetype of this phenomenon.
Yes, one does get the impression that it is something that turns on quite suddenly. It also seems like it can be turned off by hormones - that would explain why teens (especially boys) often act like animals with no self-awareness. :-)