> My dog doesn't speak any language,

If you'll allow me to interpret "speak" to include "understand", I will respectfully add a contradictory note. My dog has a vocabulary of at least dozens of words and understands them remarkably well. For example, different areas we can go to have different names and saying one of them gets her to make an immediate hard turn.

I would also argue that dogs have a gesture and body posture based language they use among themselves. They, like most other animals, are not able to make the variety of noises we humans make, so they use movement instead.

I personally can easily believe that self-awareness/consciousness and language are both near-unavoidable side effects of emergent complexity, and exist in degrees across nature.

Indeed. For some reason I can shut off my thinking instantly by holding my breath, and then I feel like a very conscious but unthinking animal.