There was a Troy in Turkey--and there probably was a war, but about in 1200 BC. Homer would have created the Iliad and the Odyssey sometime around 800 BC according to Bernard Knox's notes I'm pouring over these days.

In the introduction to Fagles' 1996 translation, Bernard Knox notes that none of the characters in the Odyssey write--none!--except one. There will be one who writes--and I haven't found that character yet in the first nine books. That's because during Homer's time hardly anyone wrote. Knox notes that the Phoenicians brought some kind of non-alphabetic 'writing' to the Greeks--and I think Faldage has commented on that alphabet along with others here at some time in some thread--but very few people actually wrote, although the Greeks turned the Phoenician symbols into a working alphabet, one sound exactly for each letter.

There's a terrific argument in Knox's introduction in which the following is proposed:

"It is not surprising that many recent scholars in the field have come to the conclusion that writing did indeed play a role in the creation of these extraordinary poems, that the phenomena characteristic of oral epic demonstrated by Parry and Lord are balanced by qualities peculiar to literary composition. They envisage a highly creative oral poet, master of the repertoire of inherited material and technique, who used the new instrument of writing to build, probably over the course of a lifetime, an epic poem on a scale beyond the imagination of his predecessors" (20).

Homer
The Odyssey
Translated by Robert Fagles
Introduction and Notes by Bernard Knox
1996
Penguin Classics

...and this translation won numerous awards in 1996, by the way.

The entire presentation of the various theories about the impossibility of the writing of the epics v. the possibility is pages long in Knox's introduction--and entirely fascinating.