Learn Swedish: Ungen sover redan. - The kid is already asleep.

Barn is the standard, neutral word for "child". Unge is more informal and translates best to "kid" (it can be used affectionately or sometimes when slightly annoyed, like "brat"). It is also the standard word for an animal's baby, like a cub or pup. Since it is an en-word (en unge), the definite "the kid" becomes ungen.

The English translation says "is asleep". Why doesn't the Swedish sentence use a form of är (is)?

Swedish doesn't use a "to be" + adjective structure for being asleep, nor does it have a continuous form like "is sleeping". You just use the standard present tense verb sover (sleeps). Therefore, ungen sover covers "the kid sleeps", "the kid is sleeping", and "the kid is asleep".

I've seen unge used as an adjective. Is this the same word?

They look identical but function differently! Ung is the adjective for "young", and unge can be its definite masculine form (like in den unge mannen - the young man). But in this sentence, unge is a noun meaning "a kid".

If I wanted to emphasize "already", could I start the sentence with redan?