Romans 10: Israel and Jesus:

Paul continues to argue why the Jews are rejecting Jesus as the Messiah, while the gentiles are accepting.
Although the Jews/Israel have zeal in obeying the Law, they are trying to get salvation instead of trusting on the Lord who gives salvation.


9-10 It’s the word of faith that welcomes God to go to work and set things right for us. This is the core of our preaching. Say the welcoming word to God—“Jesus is my Master”—embracing, body and soul, God’s work of doing in us what he did in raising Jesus from the dead. That’s it. You’re not “doing” anything; you’re simply calling out to God, trusting him to do it for you. That’s salvation. With your whole being you embrace God setting things right, and then you say it, right out loud: “God has set everything right between him and me!”

For me, it is very interesting that Paul uses the word "Israel" to talk about the Jews as in:

21 But as for Israel, He says, “I have spread out My hands all day long to a disobedient and obstinate people.”

In the church that I am attending now, they see the estate of Israel as different from the eschatological Israel. To my understanding, in my church they see the believers in Jesus as the Israel. The question is: how would Paul see the estate of Israel, the military power that we see today? Would Paul think of them as Israel in the same meaning of verse 21? If by Israel we would mean the Jews, then the estate of Israel is not composed of only Jews. Anyway, Paul is sad because Israel rejected the Messiah, but he sees it as a fulfillment of the prophecies.
 

Romans 9: Jews and Jesus

The gospel were spreading much more among the gentiles while many Jews were rejecting it. The Messiah was expected by the Jews and not by gentiles, so how come the message "Jesus is the Messiah" was so much rejected among the Jews? The Jews were descendants of Abraham while the gentiles were not chosen by the Lord. The Jews had Moses and the prophets while the gentiles have no revelation. So what was happening? Paul argues that the Lord chooses and rejects whoever he wants and Paul argues that, although the Jews were physically descendants of Abraham, they didn't trust in the Lord, they trusted on their works.
The Lord is really the lord of all universe and of all history. Paul argues that what was happening was predicted by Isaiah:

Careful! I’ve put a huge stone on the road to Mount Zion,
    a stone you can’t get around.
But the stone is me! If you’re looking for me,
    you’ll find me on the way, not in the way.

Romans 8: Trust in Jesus and get a new life powered by the Spirit

The trust/faith in Jesus set us free from the power of the flesh. It leads us to a new life powered by the Spirit. The Spirit makes us:
. please the Lord v5
. bring life and peace v6
. belong to Christ
. resurrect in a new body v11
. understand that we are children of God
. cry Aba, Father!

We read:
12 Therefore, brothers and sisters, we have an obligation—but it is not to the flesh, to live according to it. 13 For if you live according to the flesh, you will die; but if by the Spirit you put to death the misdeeds of the body, you will live.

As we belong to Christ, we have an obligation to live by the Spirit, to put to death the misdeeds of the body.

Romans 7: the Law is powerless

The church in Rome was divided in gentiles and Jews. The Jewish Tradition always gave a great importance to the Law, to the Torah; but what is the importance of the Torah in the gospel? For the Jews, the Law was good and it is crucial to obey the Law. Paul argues that although the Law is good, gentiles and Jews can't really obey the Law. Our human nature is so strong that it enslaves us. Paul writes in first person to illustrate that he too can't fight against his human nature, but he writes:

25 The answer, thank God, is that Jesus Christ can and does. He acted to set things right in this life of contradictions where I want to serve God with all my heart and mind, but am pulled by the influence of sin to do something totally different.

The Law is powerless, it can´t enable no one to obey it, but Jesus can. Thus, thank God that Jesus not only save us from our former sins but He changes our lives, breaking the power of sin that is in us.

Romans 6: united with Jesus

The believer is united with Jesus in his death and resurrection. When a Christian believes in Jesus as his Messiah, his king, his representative, he leaves the Adam (sinful) way of life, in fact, he considers his Adam sinful nature nailed on the cross with Jesus and he begins to live (resurrects) to a new life where he is slave of God and no more slave of sin. That means that the believer has power from the Lord to say "no" to temptations.

Romans 5: Peace with the Lord

What do I want more? Money? status? I am retired. Death is coming. I want to be in peace with the Lord. Sometimes I remember things that I shouldn't have done; however I trust that my sins were forgiven through Jesus. How could Jesus have died for a sinner? When I understand that the Lord has been so gracious to me, I not only become glad but also look forward to be changed in a person who loves and pleases the Lord. As we read:

1 So now, since we have been made right in God’s sight by faith in his promises, we can have real peace with him because of what Jesus Christ our Lord has done for us. 2 For because of our faith, he has brought us into this place of highest privilege where we now stand, and we confidently and joyfully look forward to actually becoming all that God has had in mind for us to be.

Romans 4: father Abraham

In Romans 3 Paul states that we are saved by faith; however, the Jews would doubt it claiming that we are declared just before the Lord by obeying the Torah. 

Paul goes to Genesis in the Torah and points to Abraham. He was declared just before the Lord because he trusted in Him, he trusted in his promises. This happened before the circumcision. Thus, Abraham is the father of those who trust in the Lord and not only those who were circumcised.

In this way, the church in Rome gathers Jews and gentiles as the people of the Lord.