Learn Italian: Il bambino assonnato sbadiglia e va a letto. - The sleepy child yawns and goes to bed.

Assonnato means "sleepy" or "drowsy" (feeling the need to sleep, related to the noun sonno). Addormentato means "asleep" (already sleeping, from the verb addormentarsi). Since the child is yawning and walking to bed, he is clearly just sleepy!

Why doesn't letto take a definite article here (like al letto)?

Andare a letto is a fixed phrase meaning "to go to bed" in the sense of going to sleep. If you said andare al letto, it would emphasize walking toward a specific physical bed (for example, to make the bed or retrieve an item from it).

Since this lesson is about adjectives used as adverbs, could I add one here and say sbadiglia forte?

Yes! Forte is one of those adjectives that can be used adverbially without changing its form. Saying Il bambino sbadiglia forte (The child yawns loudly) is perfectly natural and much more common than using the full adverb fortemente. Because it is acting as an adverb, it remains masculine singular regardless of who is yawning.