This is an excellent question worthy of a detailed answer. There are also a lot of questions here. So I'm going to break the answer down into sections for each question:
If God wants everybody to be a Christian and the Bible is the central text of Christianity, what were people supposed to do before it was completed, much less started?
Let's get some preliminary assumptions out of the way.
- What does it mean "to be a Christian"?
To be a Christian, a person must believe that Jesus, Who was God Himself, suffered the punishment for our evil (sin) by dying on the cross for our sins and rising to life again (which proved that He was, in fact, God).
- What about people who lived before Jesus? They didn't know any of this and couldn't have believed it. How were they saved?
The ancient peoples were saved in much the same way that we are saved today. As modern people, we look back in recollection at what Jesus did for us on the cross. The ancient peoples looked forward in hope of what God promised that He would do after Adam and Eve sinned.
- What did God promise Adam and Eve after they had sinned?
Immediately after Adam and Eve sinned, God offered the promise of a Messiah Who would destroy evil.
Genesis 3:15 New International Version (NIV)
"And I will put enmity between you [the Serpent/Satan] and the woman [Eve], and between your offspring and hers; he [Jesus] will crush your head, and you will strike his heel.”
After this incident, this promise from God didn't need to be immediately written down. Adam and Eve would have remembered this promise and held onto it with their every hope. They were exiled from the Garden of Eden and God's presence; they were longing every minute of every day for this promise of a Messiah to be fulfilled. With every child they had they would have remembered that one of their offspring would be the Messiah.
As their children grew up, Adam and Eve would have told their children about God's promise and the restoration of Eden. These children would have remembered the promise and would have passed it on to their children and so on. Even though it may not have been written down, these people would have been Christians (since "Christ" and "Messiah" are just different languages) since they were trusting in God's promise of a Messiah.
Is there anything on record speaking to reasons why the particular era that the Bible was written was chosen (specifically OT)?
Keep in mind that the Bible records a lot of history. This is because one of the descendants of Adam and Eve was going to be the Messiah. Adam and Eve would have wondered is Cain the Messiah? Absolutely not! Is Abel the Messiah? No. Is Seth the Messiah? No.
On and on this would have progressed. This is what makes the first genealogy in Genesis so interesting. It specifically mentions that every one of the people died. Since they died, they definitely weren't the Messiah that was promised.
No one knows who wrote Genesis. Although, most biblical scholars believe that Moses wrote the book as well as the rest of the Pentateuch. It could be that God directly gave Moses the words to write while Moses was on Sinai. God could have also indirectly written through Moses when Moses wrote down the oral traditions passed from one generation to another.
For millennia, the same story of looking for the Messiah progressed. People were longing for the Messiah and every potential candidate eventually died.
Throughout history, there were clues given about the specific lineage of the Messiah, and that was vitally important for the people living in the particular era. There are many prophecies about the Messiah, but some main ones are below.
Abraham => Genesis 12:3
Judah => Genesis 49:10
David => 2 Samuel 7:16
Each of the above passages may have led people to believe that Abraham, Judah, or David were the Messiah. But that changed when all of them grew old and died. They weren't the Messiah after all.
This is also what makes the other geneaologies throughout the rest of the Bible so important. People were looking for the Messiah and and were narrowing down the candidates based on the prophecies given above (as well as numerous others).
However, I also believe that there is another important reason why the Bible was written in the time that it was, and the answer is the same as the answer for the last question.
The Bible was gradually written. So were there parts that weren't needed until they came up, or did they perhaps correlate to certain events?
For the most part, the answer is both.
We mentioned above that God spoke to who specifically would be in the lineage of the Messiah. This would have correlated to specific events.
You could also say that some parts of the Bible weren't needed until they came up. But I would prefer to rephrase it and say that there were some parts of the Bible that people were not ready hear.
People needed to learn on their own that an ordinary human could not be the Messiah.
God chose Adam and Eve and gave them everything that they could have wanted and needed and they still failed. Unfortunately, that wasn't a fluke.
God chose Abraham, and he failed.
God chose the nation of Israel, and they failed.
God chose Moses to be His greatest prophet, and he failed (he didn't even get to enter the Promised Land).
God chose David, and he failed.
None of these people were "good enough" to destroy Satan. Because each of them were sinful and never perfectly kept the word of God, they were on Satan's side in the war against God! Because of this, they too needed a Messiah to rescue them.
So, when irrefutable proof was finally given by humanity itself that they could never fulfill the role of Messiah, God came to earth, lived the way that the Messiah should have lived by keeping all God's commandments, died in our place with the punishment that we deserved, and rose again proving that He was in fact God.
This wasn't a Plan-B effort on God's part. He knew from eternity past that humanity would never be "good enough" (i.e. perfect / sinless) to fill the role as Messiah. God knew that He would have to fill that role. But God still needed to show us that we never could have been perfect without Him.
With that, when Jesus was on earth, died, and rose to life again, the apostles wrote all their accounts (through the Holy Spirit) to finish the story. The Messiah had finally come!