tl;dr
Yes, they can be read independently.
On the nature of myth (hand-wavy background stuff)
In a comment to your question, you note:
these were originally oral traditions rather than written books.... [So] I suspect that the Odyssey doesn't require knowledge of the Iliad.
This comment is spot-on, and gets to the heart of the matter. What does "knowledge of the Iliad" mean? How does one gain knowledge about the stories told in these two epics?
We learn the stories of the Iliad and the Odyssey in the same way that we learn Bible stories. Plenty of people all over the world, including people who are not even nominally Christian, know stories that appear in the Bible: for example, that Jesus was born in a manger, or that he was crucified, or that he raised Lazarus from the dead. Reading the Bible is not a prerequisite for learning these stories. Large numbers of pre-literate children and illiterate adults know them. Prior to the Reformation, i.e., when the Bible had not yet been translated into vernacular languages and literacy was far from widespread, people nevertheless knew these stories.
That's because Bible stories are myths. Not in the sense of falsehoods, but in the sense of founding narratives a culture tells itself. Myths help a society organize itself by providing ritual and meaning. The Christian myth, for example, is supposed to not only explain the origin, purpose, and goals of human life, but also guide both the personal belief system and the communal or social behavior of adherents. The subject matter of the Iliad and the Odyssey played a similar role in ancient Greece. While Greek mythology no longer comprises the dominant belief system and organizing framework of Western society, its power as myth abides.
How do myths circulate in society? The first canto of the Mahabharata, a Hindu epic that has equivalent status in India to that of the Odyssey or the Bible in the West, provides the answer. The epic begins thus:
Sanskrit text:
नारायणं नमस्कृत्य नरं चैव नरोत्तमम्
देवीं सरस्वतीं चैव ततो जयमुदीरयेत्
iTransliteration:
naaraayaNa.m namaskR^itya nara.m chaiva narottamam
devi.m saraswati.m chaiva tato jayamudiirayet
Translation (mine and AYOR):
After bowing to Narayana and Nara the perfect man, and also to the goddess Saraswati, the Jaya must be recited.
Jaya, Victory, is an alternative title for the Mahabharata. This opening stanza specifies that the epic must be told orally, through recitation. Recitation is different from reading aloud in that it relies on memory, not on following words printed on a page.
The rest of the canto exemplifies the nature of this recitation. A young man, Ugrashravas, joins some sages in a forest. Their leader is Shaunaka. Shaunaka and his followers ask Ugrashravas where he has been. He replies that he was at a gathering where Vaishampayan recited some stories. He says that Vaishampayan had learned those stories from Vyasa, the author to whom the Mahabharata is attributed. The sages request Ugrashravas to narrate those stories to them in his turn, and he obliges. The entirety of the Mahabharata in its canonical text form embeds the act of recitation, of telling and re-telling stories from memory. Vyasa composes the verses based on his memory of events he has witnessed and participated in; Vaishampayan hears them from Vyasa; Ugrashravas hears them from Vaishampayan; Shaunaka's band hears them from Ugrashravas; the reader of the text is hearing what Vyasa told Vaishampayan told Ugrashravas told Shaunaka.
The reader of an authoritative text fixed by using the best techniques of 20th-century philology and editorial scholarship, in other words, is already at five removes from the events of the "original" story—six removes if the editorial process itself counts. This is the very nature of myth. There is no original story; all we have is stories endlessly retold, circulating so widely and dispersed so thoroughly throughout a culture that seeking to find a point of origin is entirely beside the point. The myths qua myths exist because they're constantly told and retold: as bedtime stories, as movies, as novels, as pathbreaking new translations of older works, as cartoons, as poetry.
Mythology, in other words, does not depend on an Ur-text that serves as the ultimate source of authority, and whose version of events is definitive. Even if there is a text accepted as canonical, such as the Bible, that text is a compilation, long after the fact, of various stories that have come down orally. As a result, there is no consensus about which of those stories are actually canonical. Different Jewish and Christian traditions have different canons. For example, the book of Judith is not part of the Hebrew Bible; is considered canonical in most Catholic and Orthodox traditions; and is part of the Protestant Apocrypha. And despite its non-canonicity within Judaism, the story of Judith nevertheless influences some Hanukkah observances. Lack of canonical authority does not deprive a mythological story of its force as it continues to circulate within a culture.
Further more, even when certain stories are universally accepted as canonical—for example, the Gospels of Matthew, Mark, Luke, and John among all Christian traditions—the prior processes of oral transmission and wide diffusion make it almost certain that there will be inconsistencies within the text itself. In the Gospels, for example, Mark 14: 12–16 says that the last supper was a Passover meal, while John 19: 14 says that Jesus was arrested on the afternoon before Passover. Since such inconsistencies don't impede the function of myth as a source of meaning and organization within a culture, seeking to resolve such inconsistencies is also beside the point.
Trying to construct a single, definitive, self-consistent narrative out of any given mythology would be akin to taking all the stories that have ever been told about characters in, say, the Marvel comics universe and coming up with a totality that lays out the timeline cleanly, accounts for any and all perceived inconsistencies, and retains ultimate authority. The attempt is doomed to failure. Manage it today, and someone will write a fanfic tomorrow that overturns the beautifully constructed metanarrative, ignoring the hapless cries of "But that isn't canon!" In the world of myth, everything is canon, and nothing is definitive.
On the Iliad and the Odyssey (the answer you asked for)
You've asked four questions about the Iliad and the Odyssey:
- Can the Odyssey be consumed independently of the Iliad?
- Were the original versions more independent?
- Does the Odyssey, in its original version, assume or require knowledge of the Iliad?
- Is all the necessary story background on Iliad events provided in the Odyssey itself?
Revisiting these questions in the light of this understanding of myth, one might want to examine some assumptions. Two of the questions refer to original versions of the two epics. But it makes no sense to speak of original versions of myths. One might speak of the oldest surviving written text that has come down to us, or of a transmission history that allows us to posit a text antedating extant scribal recensions, or of the first translation of a given text into a given language; none of those, however, are the original of the myth. They are all particular retellings. As a now-defunct contributor wrote in an answer to a closely related question on this site:
The Iliad and The Odyssey are a part of Greek mythology. They didn't arise from Greek mythology (any more than mythology arises from itself), because in oral tradition mythology, story[,] and history quickly blend together. [Emphasis in original].
The nature of myth also problematizes the notion that the Iliad supplies a particular background to the Odyssey. Every retelling of a myth is the background to any retelling of the myth. The Odyssey is as much part of the background to the Iliad as vice-versa. It would be rare to find a reader unacquainted with certain basic information about Greek mythology:
- Helen, a Greek queen, was the world's most beautiful woman
- She ran away with, or was kidnapped by, Paris, a prince of Troy
- All the Greek kings waged a long war against Troy to get Helen back
- Ultimately, because the Trojans failed to look a gift horse in the mouth, the Greeks won
- Afterwards, one of the Greek heroes took a really, really long time to get home from Troy because he kept being waylaid by the most incredible adventures.
All the two epics do is fill in the details to this basic information, with many ancillary stories. Even the Iliad doesn't provide background information in the sense of a cogent, linear account of what led up to the war; it begins, as epics do, in medias res. A hypothetical reader who comes to the Iliad with no prior background in Greek mythology would be utterly baffled about why these dudes are duking it out in the first place. We can discount such a hypothetical reader, however; someone wanting to read the Iliad is ipso facto likely to be familiar with Greek mythology to begin with.
The Iliad doesn't end with any sort of finality either. It ends with the burial of Hector, one of the Trojan heroes. The war is still unresolved at the end of the narrative. As a consequence, someone coming to the Odyssey with just the knowledge from the Iliad, having read the latter as "background" and with no other information about the story of the Trojan war, would be equally baffled about what's going on. Wait, Telemachus is visiting Helen and Menelaus in their palace? She's back in Greece? When did that happen? Did the Greeks win? How? ¡¿A mfing wooden horse?!
The inconsistencies between the two narratives also show that the events of the Iliad do not flow seamlessly into those of the Odyssey. A blog post on The Masculine Epic notes:
The lives of Aphrodite and Hephaestus are also inconsistent between the two poems. In the first, Hephaestus is married to a minor goddess, the Grace Charis. In the latter, he is Aphrodite’s cuckolded husband.
As has been argued above, such inconsistencies are taken for granted within a mythological universe.
Given their circulation as myth, making sense of the narratives of the Iliad and the Odyssey depends heavily on prior knowledge of the stories they retell. The epics amplify and elaborate upon events that are always already known to the reader, albeit not in a unitary or definitive form. The retelling gives this diffuse knowledge a particular expression or shapes it into a particular artifact. That expression or artifact itself goes on to become part of the generalized diffusion of the myth in culture.
Every version of a myth assumes a reader who is familiar with other versions of the myth. Not necessarily in any scholarly, comprehensive, or definitive way, but just by virtue of being part of a culture that has already widely recirculated the myth and continues to do so. This being the case, to call the Odyssey "largely a sequel to the Iliad", as the question does, mischaracterizes both those epics. The one isn't anterior to the other, and doesn't provide information that's straightforwardly necessary and useful for reading it. The Iliad merely tells a story whose action occurs prior to the action of the Odyssey. But that doesn't make the Odyssey in any way dependent on the Iliad. It's perfectly fine to eat on its own.
