Learn Icelandic: Kötturinn hennar er mjög fallegur.

In Icelandic, when you use a possessive word like hennar (her), the noun it belongs to usually takes the definite article (the "the" ending). So instead of just "cat her", you literally say "the-cat her" (kötturinn hennar).

Does hennar change if she owns a feminine word or multiple things?

No! Hennar is an invariable genitive pronoun meaning "of her". Because it just states who the owner is, it stays exactly the same no matter what she owns. It will always be hennar whether the item is masculine, feminine, neuter, singular, or plural.

Can I put hennar before the noun, like we do with "her" in English?

You might see that in older texts or poetry (which would be hennar köttur without the "the" ending), but in everyday spoken Icelandic, the possessive almost always goes directly after the noun: kötturinn hennar.

Why does fallegur end in -ur here?

Fallegur (beautiful) is an adjective, and in Icelandic, adjectives must match the gender of the noun they describe. Since köttur (cat) is a masculine word, the adjective takes the masculine singular ending -ur.