Learn Icelandic: Sérð þú barnið þeirra?

In Icelandic, possessive words usually come after the noun. When they do, the noun must have the definite article (the "the" ending) attached to it. So, "their child" is built as "the child theirs" (barnið þeirra).

Does þeirra change if they own a different word, like a masculine noun or plural nouns?

No, þeirra is invariable. It is literally the genitive (possessive) form of the pronoun for "they", meaning "of them". Because it just tells us who the owner is, it doesn't change its spelling to match the gender, number, or case of the thing being owned. It stays þeirra no matter what they own.

Why does the sentence start with Sérð instead of using a word for "Do"?

Icelandic doesn't use helper verbs like "do" to form questions. To ask a yes/no question, you simply swap the subject and the verb. Changing Þú sérð (You see) to Sérð þú makes it a question: "Do you see?".

What grammatical case is barnið in here?

It is in the accusative case because it is the direct object receiving the action of the verb sjá (to see). However, because barn is a neuter noun, its accusative form looks exactly the same as its basic nominative (dictionary) form.