At the time of this posting, this question has 4 close votes. Several comments have leveled accusations of being off-topic as not about Jewish Life and Learning.
However, the question asks "According to Jewish thought, do animals have the self-awareness to know that they exist?..."
I am having trouble fathoming why this is not about Jewish Learning. I am further having trouble understanding the comments that were bothered about the lack of practical ramifications, while questions about if angels have free will, or if angels have jealousy, remain untouched by such complaints. In fact, skimming the hashkafa tag, I see too many questions that do not have practical ramifications to list. I don't see this as a problem.
Another comment criticized the question for being an "open ended philosophical discussion, not necessarily answerable." I don't know why it is any more open ended than any other philosophical question. And every question that asks "is there a source for x" is "not necessarily answerable" if the standard of "answerable" means there is someone somewhere who says what you are looking for. There is certainly the possibility of an answer, and I don't know why there shouldn't be Jewish sources that discuss it.
I think the question could be informative in understanding several things, among them how we should relate to the prohibition of tza'ar baalei chaim (are animals just mobile chunks of flesh, or do they have a conscious and possibly experience pain?). But even if that wouldn't be the case, I think understanding the nature of G-d's world in a vacuum is enough of a value.
Can anyone explain what is wrong with this question?