I don't think the terms "self-aware" and "conscious" are defined well enough to give you an answer. On one hand, we haven't given the title "self aware" to anything smaller than an Orangutang. However, depending on your definitions, you could make a hard sell that a standing wave such as a quantum waveform has self-aware traits.
My answer, if I had to peg definitions for "self-aware" and "conscious," would be that the answer depends on how hostile the environment is. The more hostile the environment is, the less value there is in understanding yourself (you spend all your energy battling the environment). The friendlier the environment is, the more an understanding of how "you" operate becomes an important trait.
At the smallest scales, Brownian motion is a dominant force. Its randomness is going to keep a small organism evolving to deal with its effects, not its effects on itself. However, if temperatures were chilled, it may evolve to be more aware of itsself on its surroundings.
Pack your local area close enough, however, and your neighbors all work like you. Suddenly its really useful to understand yourself so you can understand your neighbors.
Bacteria exhibit interesting behavior in biofilms. They effectively form a quorum, at which point they all agree that banding together is good for the group. They then react differently, "realizing" the power of their combined force is greater than the sum of their Selves.
What about parts of a body? Currently there is belief that the Anterior cingulate cortex may possibly be the "throne" of the brain's consciousness. Is it? We don't yet know. But this does suggest that there's a smaller part of our brain responsible for this funny thing we call consciousness.
Going all the way to the waveform argument, consider a standing wave in a flute. If any air molecule is perturbed, such as from Brownian motion, it falls back in sync, because it "knows" the relationship between its momentum and its position. Or does it? This quickly becomes a philosophy question. What does it mean "to be aware of one's self?"
Philosophy majors will take it all the way. There is a category of philosophies called idealism which all center around the idea that everything is conscious thought. Every grain that makes up the sandy beach is a conscious aware entity. It's a bit extreme, but it points out just how hard of a question you are asking. In the other direction, there is a physicalilst argument that consciousness is simply an illusion, and nothing is conscious, not even us.
Edit: in light of your edit, I recommend two Wikipedia articles:
Animal consciousness is not a well defined topic. Some tests, such as the mirror test, admit some primates and pigs, but dogs often fail that test (which is counter intuitive for many dog owners). On the other hand, some tests are with regard to the handling of pain, which many animals, including dogs, pass.
The actual issue you are going to have to grapple with is the Problem of Other Minds. It is an extremely difficult topic, which can go so far as to say "all other humans are merely zombies... I am the only conscious one." I recommend looking into this problem, because your edit suggests this direction: you are seeking to make them "similar" enough to humans to ease identification via the ability to vocalize and so forth.
Because of how hard the Problem of Other Minds is, and how much harder it is to convey this issue as an author, consider side-stepping the issue entirely. Use Sanderson's First Law of Magic to make your life simpler:
Sanderson’s First Law of Magics: An author’s ability to solve conflict
with magic is DIRECTLY PROPORTIONAL to how well the reader understands
Decide to yourself "this creature size is conscious." It doesn't really matter what line you pick, pick a line that suits you and your intuition. Call it magic -- you defined something magical called consciousness. Cool! Now by Sanderson's first law, you are free to call them "conscious" as long as you please, up until the point where you try to resolve conflict with it. At that time, you need to have taught the reader enough about this "micro-consciousness" that the reader understands enough to let you solve the conflict.
Because the understanding of your particular definition of consciousness is narrow at the start of the work, concentrate on showing how it solves problems that the reader would relate to that sized animal. For instance, ants don't like water on their nest. It does really bad things when the nests flood, so they have to take action like move the young closer to the entrance. An ant would be interested in keeping the nest area dry. However, they would not invent robotics to cut down a giant banana leaf to cover the nest, but they would develop gorgeous Mandela of small leaves which interlock to make something watertight, but which can be moved by a single ant. Now we have demonstrated problem solving.
Now lets say there's a particular parasite of a larger creature that is considered "good food" by the ants (for any reason you please). Every now and then, an ant might come across one and get to take it back, but its a lot of work. However, the larger creatures are smart -- they can be trained. Consider an ant colony that will rearrange their Mandela to signal nearby creatures with parasites to come be groomed (or they may arrange their Mandela differently to signify that they have enough food, and would prefer to leave their nest less-trampled). This would be along the lines of the small cleaner wrasse who clean the mouths of sharks. Suddenly we have the ant colony communicating with a greater species, and communication is a strong "walks like a duck" moment in any consciousness story.
At some point you should be able to wrap the reader up into questioning "is the ant conscious, or is the ant colony as a whole conscious?" That is the point where you have them wrapped up in your particular definition of consciousness, and you can start saying "let me explain to you why this is the sound of a duck quacking," and no reader will disagree.
The key to all of this is that you never use consciousness to solve a conflict that the reader does not already understand: and you start from conflicts that could be reasonably resolved by a creature of similar size and move up.