Interestingly, the Greek never indicates explicitly that John is "the one who Jesus loved." Though it is certainly implied by the fact that John is never explicitly named in the gospel while almost all the rest of the disciples are. The Greek uses the term ἠγάπα from the verb ἀγαπάω. Greeks had their choice of love verbs. This is decidedly Platonic, in that it is explicitly distinct from ἐράω (sexual love), but also has a different shade of meaning from φιλέω (affectionate love). ἀγαπάω has an implication of respect and mutual regard and was the verb of choice for the relationship between God and man in Greek translations of the tanakh.
Now the question of familiarity and authorship gets into some tricky waters and I will start with the caveat that, although I know my Greek, Biblical history is not my specialty. As far as I understand it though, it is unlikely that the gospel of John was written by the disciple John. For one thing, it seems to have been the last of the four canonical gospels and expected the audience's familiarity with Mark and Luke. It is usually dated to sometime in the 90s, so John, if alive, would have been rather old. It seems that the followers of different disciples created different texts based on the teachings of their leaders and an official Christian canon was not decided until the third century or so.
There were at this period a number of different gospels floating around and not all of them made the cut (for example, check out the weird gospel of thomas). Referring to John within the gospel as one defined by a special relationship with Jesus may have been one way that this sect promoted the truth of their gospel.
I hope this helps a little with your question, though since this is not my specialty I know I'm generalizing and might be a little off in my emphasis. For more on the authorship of John, you might want to look at John Davidson's The Gospel of Jesus: In Search of His Original Teachings
On “the disciple whom Jesus loved”
As for the translation, no – not really. “the one who(m) Jesus loved” renders the Greek quite straightforwardly. Yes, there are different words for ‘love’ in Greek, but the suggestion that in New Testament Greek they can be strictly divided into words with clear-cut distinctions is not linguistically sound at all. Yes, agapē is often used for divine love, but it’s also used in the Septuagint to describe incestuous desire that leads to rape (2 Sam 13:15).
Dating and Audience
Generally it’s held that John’s Gospel was written late, 90-110 AD, though some would push it back even into the 2nd century. It’s generally (again, that word!) held to be the product (and addressed internally to) a Jewish-Christian community which has been excluded from the synagogue(s). They would not have been familiar with Jesus directly. In any case, given the late dating of the gospel, the faith had already spread well beyond that. It’s therefore a temporal issue, not a geographical one necessarily.
Authorship of John
Strictly speaking John, like the other canonical gospels, i anonymous. The (4) Gospels all carry an ascription added later. It does, as you note, have a figure mentioned throughout as ‘the disciple whom Jesus loved’. And 21:20 mentions this disciple before 21:24 says:
This is the disciple who is testifying to these things and has written them, and we know that his testimony is true. (NRSV)
So the text makes an internal claim that the beloved disciple is the source of the gospel, if not its author. This, coupled with the fact that John the Apostle frequently appears in the ‘core’ of the Twelve in the Synoptics, and that John and James are not mentioned (except obliquely in 21.2) in this gospel, suggests John the apostle as the author. This has been the traditional position.
Bauckham has argued that the BD is indeed the author, but that this should be identified as John the Elder, not John the Apostle. His argument is that the BD is formally anonymous, but not part of the Twelve apostles. Just close by! Based on the fragment of Papias preserved in Eusebius’ Ecclesiastical History 4.14.7, there is this second John figure, John the elder, who should be regarded as author.
Witherington has argued for the noel position that Lazarus might be the author. I won’t review that argument here. Other proposals have included Mary Magdalene, Thomas, and the Samaritan woman. And then there are the theories that it is a composite text, either authored by an individual and then redacted several times over, or entirely a composition of pastiche parts.
On the balance, contemporary critical scholarship does not hold that John the Apostle = BD = author of the gospel, but even so, continues to suggest that the BD is the source behind the gospel, written then by a truly anonymous figure.