Consciousness without language
Language is a means of expression and communication. It serves as a medium of expression. Consciousness is independent of language. One (human or any animal) can be conscious however to report and verify, as you said, we need a language or any medium of communication.
Consciousness could safely considered an axiom in psychology.
For the second question, you have bumped on to one of the canonical pairs. How do we explain nothing? Ideally we cannot. The same is with consciousness. It is a fundamental thing. A formal verification might require a revisit to the fundamentals of philosophy and psychology.
Reference:
Thought without language: Thought without awareness? - L.Weiskrantz