I thought I would give a "Tale of Two Old Testaments" by posting the version Paul might say with historical footnotes. What do you think?
God created the world in such a way that his eternal power and divinity should have been apparent to everyone, even now (Rom. 1).  However, when Adam sinned, the power of Sin entered the world. The creation became a slave to corruption (Rom. 8). Death entered the human race because of Adam's sin (1 Cor. 15). We all sin now like Adam did and we all die now like Adam did because of the power of Sin over the world (Rom. 5). 
God called Abraham to begin the salvation of the world. Abraham is the father of the circumcised and the uncircumcised, of the Jew and the Gentile (Rom. 4). Before he was circumcised, Abraham is the father of the Gentiles who become right with God on the basis of their faith in him. After he was circumcised, he is the father of Israel who remains true Israel when they have faith in God. Their "works of Law," meaning the things they do that separate them from Gentiles (circumcision, food laws, sabbath observance) cannot earn them a right status before God on the Day of Judgment.
Great honor is due the Jews because God gave them the covenants, the temple, the promises (Rom. 9). The revelations of the Scriptures were given to them (Rom. 3). But none of these things in themselves are sufficient to give them a right status with God on the Day of Judgment, which is a matter of God's grace.
The Scriptures witness to these truths, both the Law and the Prophets (Rom. 3).  From Habakkuk 2:4 we learn that "the person who is righteous on the basis of faith will live." Joel 2 tells us that whoever calls on the name of the Lord (Jesus) will be saved.  We've already mentioned key texts about Abraham in Genesis.
God gave the Law to be a guardian to point Jews to Christ (Gal. 4), like a slave that watches over a child until it comes of age. Israel was under that guardian until Christ came. The Law told Israel what God's righteous standard was, basically love. But Jews could not keep that standard because they did not have the Spirit. 
David is significant because he began the royal line that would lead to the Christ. Jesus is the seed of David in fleshly terms. Scriptures foresaw that he would become Son of God also, passages like Psalm 110:1.
Other parts of the New Testament emphasize parts of the Old Testament that Paul does not so much focus on. For example, the Gospels locate the ministries of Jesus and John the Baptist against the backdrop of the exile. John the Baptist is like the voice of one calling the wilderness to come home from the Babylonian captivity (Isaiah 40), although John is preparing the way for Jesus.
The very idea of the kingdom of God and the gospel news presumably come from Isaiah 52, which was also originally about the return from exile. Isaiah 53 shows up in Acts 8 as a text foretelling the redemptive suffering of Jesus. Matthew 2 alludes to the exodus in relation to one event in the life of Jesus and, like Joshua, Jesus does save his people. He is the Passover lamb (1 Cor. 5). He is a priest to end all priests (Heb. 7).
 Given what Jews likely believed at the time, Paul probably would have assumed that the world was created out of chaotic, pre-existing waters. But he doesn't say this anywhere.
 This is an expansion of the Genesis story. In Genesis, Adam and Eve would have died anyway if they had not had access to the Tree of Life. In that sense, death was not caused by Adam's sin but it was not prevented because of Adam's sin. The ground is cursed because of Adam's sin in Genesis, but there is no mention of the creation being corrupted because of it.
The curse of Adam plays no role in the rest of the Old Testament. In inductive terms, Genesis 4 through Malachi say nothing of this sin or its consequences. If we did not have Romans 5 and 1 Corinthians 15, Christian theology would look significantly different in relation to Adam.
 Interpreted spiritually rather than literally.
 Originally a reference to YHWH, now strikingly applied to Jesus.
 This seems to be a rather unique perspective for a Jew to have. The default Jewish understanding was probably that it was quite possible to keep the Law to the standard God expected despite the fact that all humans had sin and needed God's grace and atonement in general.