Learn Swedish: Varför ser barnet så sorgset ut? - Why does the child look so sad?

The base form of the adjective is sorgsen. Normally, to make an adjective neuter to match a neuter noun like ett barn, you just add a -t. However, for adjectives ending in an unstressed -en, you replace the -n with a -t. Therefore, sorgsen becomes sorgset rather than sorgsent.

Since barnet (the child) is definite, shouldn't the adjective end in "-a"?

No, because the adjective comes after the verb here. It describes the noun indirectly as a predicative adjective. In Swedish, adjectives only take the definite -a ending when they come directly before a definite noun, as in det sorgsna barnet (the sad child). After the verb, they only agree in gender and number, which is why we use the neuter singular form sorgset.

Why is the word ut at the very end of the sentence?

It pairs with the verb ser to form the phrasal verb ser ut, which means "looks" or "appears" in terms of physical appearance. Swedish phrasal verbs are often split up by the subject and other words, which is why ut lands at the end of the question here.

Why does the verb come before the subject (ser barnet)?