Learn Spanish (Spain): Mi hermana necesita un hacha afilada para cortar el árbol. - My sister needs a sharp axe to cut the tree.

Because hacha starts with a stressed 'a' sound (the 'h' is silent). To avoid the awkward clashing sound of saying una hacha, Spanish borrows the masculine article un for the singular.

If it uses un, why is the adjective feminine (afilada instead of afilado)?

The noun hacha is still fundamentally feminine! It only borrows the masculine article to fix a pronunciation issue. Adjectives describing the noun must always match its true gender, which is why we use afilada.

What would happen to the words un hacha if my sister needed multiple axes?

If it were plural, the 's' in the plural article breaks up the clashing 'a' sounds. The pronunciation problem disappears, so you go back to the regular feminine article: unas hachas afiladas.

Why do we use para instead of por before cortar?

When followed by an infinitive verb like cortar (to cut), para is used to express a purpose, goal, or intention. You can think of para cortar as translating exactly to 'in order to cut'.