What were people supposed to do with an incomplete or nonexistent Bible?

The word "incomplete" in your question implies that the Bible today is the complete word of God, that we have a closed canon. I've written about that elsewhere; it's really bad theology, and a concept that historically has always been associated with apostate groups, not with God's true disciples.

Once you understand that, the rest of the question is simple to answer, by breaking it down into its parts:

• We turn to the Bible today to find God's word and his will for us.
• The writings in the Bible that contain God's word and his will were given by prophets and apostles, with new teachings and concepts being given when they were needed.
• So where did people turn in ancient times when they needed to know God's will and existing scriptures did not suffice? To the currently-living prophets and apostles, of course.

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.

1. 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).

2. 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.

3. 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!

Though the Bible was nonexistent immediately after Jesus died, and it was decades before all the books we now have were written, the early church was following the teachings of Christ, as taught not only by Jesus to the Apostles, but also passed on by other disciples of Christ.

The idea of only using the Bible as the source of Christ's teachings is not supported by the Bible.

For example, the teachings that were passed on orally were supported here:

2 Thessalonians 2:15 New International Version (NIV) 15 So then, brothers and sisters, stand firm and hold fast to the teachings[a] we passed on to you, whether by word of mouth or by letter.

All Scripture does not mean just the Bible, as the Bible wasn't written yet, though some parts were, so this must also mean the oral teachings, which the Roman Catholic Church refers to as Sacred Tradition.

2 Timothy 3:15-17 New International Version (NIV) 15 and how from infancy you have known the Holy Scriptures, which are able to make you wise for salvation through faith in Christ Jesus. 16 All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness, 17 so that the servant of God[a] may be thoroughly equipped for every good work.

And for more on whether we should follow Sacred Tradition or Sacred Scripture or both: http://www.cfpeople.org/Apologetics/page51a031.html

What is interesting is that the early church had many writings to refer to, many of which we don't include as being inspired, but it is assumed that many people felt they were worth keeping, as they survived over the centuries.

For a good list of these books you can look at http://en.wikipedia.org/wiki/New_Testament_apocrypha.

Also, I don't know if God wants everyone to be Christian, but for all to have a relationship with Him and with others, as He commanded. If we focus more on religion and listen to what our ministers tell us, instead of listening to God and following His commandments, then any miracles they may have done, anything they did to appear righteous is for naught. You can look at 1 Cor 13 and this:

Matthew 7:21-23
New International Version (NIV)

21 “Not everyone who says to me, ‘Lord, Lord,’ will enter the kingdom of heaven, but only the one who does the will of my Father who is in heaven. 22 Many will say to me on that day, ‘Lord, Lord, did we not prophesy in your name and in your name drive out demons and in your name perform many miracles?’ 23 Then I will tell them plainly, ‘I never knew you. Away from me, you evildoers!’

The entire old testament was available and written before Jesus' time. The new testament, as we have it is the collected writings of the apostles. Before all the writings were complete, the apostles taught directly. After their death, their immediate successors taught what they had learned and during that time the writings were collected, copied and distributed.

Creeds and hymns formed a big part of conveying biblical doctrine before a printed bible was widely available to all, as did the church teaching and structure. The documents themselves were available to clergy from very early times, soon after the apostolic day.

So at first, there was the testimony of direct witnesses. Then there was teaching from those taught by the apostles and the apostles written records. Finally those records were compiled into the bible, and between the mid 300s and late 600s AD the canon of authoritative scripture was considered closed by most traditions, although the Catholic church did not formally declare the bible canons closed until the council of Trent in 1546 with the only disagreement with other traditions being the OT deuterocanonical books (and bear in mind, too, that the Catholic church (and perhaps others), holds to a continued revelation through the church's magisterial office and authority).