Learn Russian: Портрет висит напротив зеркала.

The preposition напротив (opposite) requires the Genitive case. Since зеркало is a neuter noun, its Genitive singular form changes the ending from -о to -а, making it зеркала.

Why use висит here instead of вешает for "hangs"?

Висит is from the verb висеть, which describes the physical state of something already hanging. You use this when the item itself is the subject. Вешать is the active process of a person hanging something up, which takes a direct object.

Could I use перед instead of напротив to say the portrait is in front of the mirror?

They have slightly different meanings. Напротив specifically means "opposite" or "facing," implying they are situated across a space from each other. Перед just means "in front of" and would require a different case, the Instrumental (перед зеркалом).

Where does the stress fall on the word напротив?

The stress is on the second syllable: на-ПРО-тив. It helps to remember that it contains the root word против (against), which keeps its original stress.