Learn Russian: Он живёт напротив парка.

The preposition напротив (opposite / across from) always requires the noun following it to be in the Genitive case. Since парк is a masculine noun ending in a consonant, it takes the ending -а in the Genitive.

I know the verb for "to live" is жить. Where did the "в" in живёт come from?

The verb жить has a slightly irregular present-tense stem. When you conjugate it, a -в- is added before the endings: я живу, ты живёшь, он живёт, and so on.

Do I always have to state what is opposite, or can I just use напротив on its own?

You can absolutely use напротив on its own as an adverb! If the context is already clear, you could just say Он живёт напротив (He lives across the street).

Does the letter ё in живёт affect where the word is stressed?

Yes! In Russian, the letter ё is always stressed. Whenever you see it in a word, you immediately know to put the emphasis on that syllable.