Learn Swedish: Hon är barnet.

In Swedish, "the" is attached to the end of the noun instead of being a separate word before it. Because barn (child) is an "ett-word", we add the suffix -et to the end to make it barnet (the child).

Does the ending of barnet change because the sentence is about a girl (hon)?

No! The word barn is always an "ett-word" and becomes barnet in the definite form ("the child"), regardless of whether the child is a girl (hon) or a boy (han). Grammatical noun gender doesn't change based on who the person actually is.

Since barn is an ett-word, why does the sentence use hon instead of det?

While barn (child) is grammatically an ett-word, we use the natural pronoun hon (she) when we are talking about a specific girl. We would generally use det (it) if the gender is unknown or when making a generic statement like "it is a child".

Is the word är pronounced exactly the way it is spelled?

In very formal situations, yes, but in everyday spoken Swedish, är is very often shortened. You will frequently hear it pronounced so it sounds almost exactly like the English letter "e".