It does not refer to any religious organization. Look up the scriptures under the dates of certain doctrines. Agapao Love is Divine Love without condition for its giving.
Phileo love is the love for a friend or brother.
On the last question, Jesus uses Phileo and Peter responds with the SAME verb. History does not give any evidence whatsoever of the presence of Peter in Rome. Scripture has no mention of Peter being in Rome. First, we need to look at what is taught about what we need to be saved FROM, according to the church of Rome. The church of Rome has separated sin into these two classes, each with their own consequences and remedies.