Learn Italian: Ha invitato tutti, perfino i vicini. - He invited everyone, even the neighbours.

While anche means also or too, perfino (and its exact synonym persino) is stronger and adds a sense of surprise or extremity. It highlights that inviting the neighbors was an unexpected action, capturing the emphatic even in English.

Could I use addirittura in place of perfino?

Yes, addirittura is a perfect alternative here. Both words emphasize the unexpected nature of the action. Saying addirittura i vicini would carry the exact same meaning of even the neighbors!

Why doesn't tutti have an article after it in this sentence?

When tutti acts as a pronoun meaning everyone, it stands completely on its own. You only need the definite article when tutti acts as an adjective describing a specific noun, such as in tutti i vicini (all the neighbors).

How do we know the subject is he if there is no pronoun?

Out of context, we actually do not! Because Italian naturally drops subject pronouns, ha invitato could just as easily mean she invited or you (formal) invited. The English translation provides he as a default, but in a real conversation, the context would make it clear.