In short Paul's intended audience in his letter to the Romans:
1:1-1:15 - Greeting the church in Rome
1:16-11:10 - Addressing Jewish Believers and Jewish objections to the Gospel to the Gentiles.
11:10-11:32 - Addresses Gentile Believers.
11:33-11:36 - Praises to God
12:1-15:21 - Practical address to all believers
15:22-16:24 - The church in Rome.
16:25 - End - Praises to God
Paul intention is to make Rome a place from which he can reach Spain:
Whensoever I take my journey into Spain, I will come to you: for I trust to see you in my journey, and to be brought on my way thitherward by you, if first I be somewhat filled with your company...When therefore I have performed this, and have sealed to them this fruit, I will come by you into Spain.(Rom 15:24&28)
The problem is that the Church in Rome must have primarily consisted of Jewish Christians. Paul did not start the Church in Rome, it must have been started by those Jews and Proselytes that were present at Pentecost (Acts 2:10) and the persecuted Christian Jews that fled after Saul started making havoc in the Church (Act 8:4) and they only preached to Jews (Act 11:19).
It is very evident from Acts that Paul and his Gospel to the Gentiles were misunderstood and rejected by the Jews, even the Christian Jews.
And when they heard it, they glorified the Lord, and said unto him, Thou seest, brother, how many thousands of Jews there are which believe; and they are all zealous of the law: And they are informed of thee, that thou teachest all the Jews which are among the Gentiles to forsake Moses, saying that they ought not to circumcise their children, neither to walk after the customs. What is it therefore? the multitude must needs come together: for they will hear that thou art come. Do therefore this that we say to thee: We have four men which have a vow on them; Them take, and purify thyself with them, and be at charges with them, that they may shave their heads: and all may know that those things, whereof they were informed concerning thee, are nothing; but that thou thyself also walkest orderly, and keepest the law. As touching the Gentiles which believe, we have written and concluded that they observe no such thing, save only that they keep themselves from things offered to idols, and from blood, and from strangled, and from fornication. (Act 21:20-25)
Notice that James, is convinced that Paul did not forsake the ordinances in the Law given to Moses, but testifies that the Jews (even those which believe) thinks that Paul has and teaches other Jews to do the same.
The persecution that Paul was facing was not for the same reason as Peter and John, who only suffered at the hand of the Authorities, but the multitude gave them an audience. They gave Paul an audience up until he brings Gentiles into the Gospel:
Men, brethren, and fathers, hear ye my defence which I make now unto you. (And when they heard that he spake in the Hebrew tongue to them, they kept the more silence: and he saith,...And I answered, Who art thou, Lord? And he said unto me, I am Jesus of Nazareth, whom thou persecutest. ...And he said unto me, Depart: for I will send thee far hence unto the Gentiles. And they gave him audience unto this word, and then lifted up their voices, and said, Away with such a fellow from the earth: for it is not fit that he should live. (Act 22:1-2,8,21-22)
Notice that the Jews had patience with the Gospel of Christ, but not with it to include the Gentiles!
Paul had been battling this opposition everywhere he went, and it hindered his ministry greatly. The biggest problem was that they would not give him an audience. If the Church in Rome did consist primarily of Jews who had the same affections against Paul and his ministry, he would not have been able to effectively preach the Gospel in and from Rome because they might have employed similar tactics that they did in other places. The letter appears to me to be his most sincere effort to get them to hear his Gospel and prove from the Scriptures that it included the Gentiles without expecting them to become Jews.
In Chapter 1 he introduces the theme of his letter:
So, as much as in me is, I am ready to preach the gospel to you that are at Rome also. For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek. For therein is the righteousness of God revealed from faith to faith: as it is written, The just shall live by faith. (Rom 1:15-17)
He proceeds to paint a picture of what appears to be "Gentiles", that, "though they knew God worshipped him not as God" and as a result has been given "over to vile affections".
You cannot help but get the idea that the "righteous" person hearing this will start bobbing his head in agreement of the condemnation pronounced over these wicked people and their wicked acts, but then frowning at the last verse:
Who knowing the judgment of God, that they which commit such things are worthy of death, not only do the same, but have pleasure in them that do them. (Rom 1:32)
And then totally caught of guard by Paul's intention of getting them to agree to this condemnation:
"Therefore thou art inexcusable, O man, whosoever thou art that judgest: for wherein thou judgest another, thou condemnest thyself; for thou that judgest doest the same things. But we are sure that the judgment of God is according to truth against them which commit such things. And thinkest thou this, O man, that judgest them which do such things, and doest the same, that thou shalt escape the judgment of God?" (Rom 2:1-3)
He proceeds in Chapter 2 to demonstrate that the Jew sins yet that feels justified in hearing the Law and fulfilling certain ordinances, is no different than the Gentile who is base altogether. While, if a Gentile fulfilled the righteousness of the Law, then he would be altogether as a Jew who kept the ordinances:
For circumcision verily profiteth, if thou keep the law: but if thou be a breaker of the law, thy circumcision is made uncircumcision. Therefore if the uncircumcision keep the righteousness of the law, shall not his uncircumcision be counted for circumcision? And shall not uncircumcision which is by nature, if it fulfil the law, judge thee, who by the letter and circumcision dost transgress the law? For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh: But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God. (Rom 2:25-29)
Paul continues to prove that the problem is sin - breaking the righteousness of the Law - the witness to the righteousness of God and that fulfilling certain ceremonial practices does not save anyone. Everyone has sinned and in God's wisdom and demonstration of His righteousness, he gave Christ, His Son, to be a propitiation to all that would believe in Him to justify them from their past sins, completely apart from the Law but exactly because of it.
You see, the Jewish sacrifices for atonement only applied to the Jews, and to be a Jew you had to have circumcision, observe dietary restrictions and celebrate the Sabbath Feast to partake in the sacrifices - or "bear your own iniquity". Which is why the Jewish Christians wanted to insist that Gentile converts become Jews through circumcision so that the Sacrifice of Christ might apply to them. Paul argues, no... the Sacrifice of Christ is for all that believe.
In Chapter 4 he demonstrates that this has always been God's plan of salvation, and even Circumcision was the sign/seal of Faith and righteousness not the source of it.
In Chapter 5 he then explains that the justified believer is delivered into the love of God that not just justified him by the death of His Son while he was an enemy, but now much more will save him through his life. In the same way, that sin and its powers (death) were imputed even before the Law was given, so righteousness and life of Christ can be imputed without the Law through faith. So the Law's purpose was not to ultimately to impute sin or righteousness but "that the offence might abound. But where sin abounded, grace did much more abound".
All throughout, Paul is addressing the objections that he has been facing over the years from Jewish objectors and in Chapter 6 its the charge that that would imply that we can now sin to experience more grace. To which Paul answers emphatically, NO! In Christ we are dead to sin: What fruit had ye then in those things whereof ye are now ashamed? for the end of those things is death. But now being made free from sin, and become servants to God, ye have your fruit unto holiness, and the end everlasting life. For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord. (Rom 6:21-23)
In Chapter 7 we are said now to be dead to the Law and alive/married unto Christ, and he is addressing the objection that "What? Is the Law sin?". For Paul, the Law taught him what sin is, and that the "Law and commandment is good, holy and just". He even "delights in the Law in his inward man"... but sin IN ME is the problem and the Law is utterly unable (nor was it meant) and unhelpful to deliver me from sin in my flesh. The problem even for Paul, is not the Law of God which he serves with his mind, but the Law of Sin which he serves with his flesh - making him unable to do good and not evil. "Who will deliver me from the body of this death? I thank God, through Christ Jesus my Lord"
Then follows the glorious chapter 8! How did God deliver us from this body of death: There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit. For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death. For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh: That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit. (Rom 8:1-4)
The Spirit not just frees us from the law of sin and death, empowering us to fulfil the righteousness of the Law, but also affirms us as children of God, prays for us and finally assures us and ensures that those who are justified, were foreknown, predestined, called and glorified. Nothing can change that!
Paul was no doubt labeled an "anti-Semite" but in the next few chapters, he emphasizes his passion for Israel - willing to give up his own salvation if it meant they could be saved. But it's not your genes that make you a child of Abraham (a child of the promise), its the sovereign election of God, as is demonstrated in Isaac and Ismael, Jacob and Esau, "that he might make known the riches of his glory on the vessels of mercy, which he had afore prepared unto glory, even (in other words) us, whom he hath called, not of the Jews only, but also of the Gentiles?" (Rom 9:24)
"Brethren, my heart's desire and prayer to God for Israel is, that they might be saved."(Rom 10:1) The problem is they seek it through the Law and not through faith in Christ and therefore "have not submitted themselves unto the righteousness of God." He proves from Scripture here (and has been all throughout) that this message has always been intended for all people, because "whoever shall call upon the Name of the Lord shall be saved". But now how can they (the Gentiles) call on him who they have not believed, how can they believe on him whom they have not heard and how shall they hear without a preacher?" Enter Paul.
It was all a glorious defence of the full Gospel message and his calling as the Apostle to the Gentiles!
Israel rejected this Gospel, but even that was all part of God's plan, in order that it may go out into all the nations, and it will continue to do so until the fullness of the Gentiles comes in. (Chp 11)
That is where Paul starts addressing the Gentiles and admonish them not to grow arrogant against the Jews because the reason why many Jews are at this time excluded is unbelief, and Gentiles are included because of faith... therefore be not high minded. It was part of God's plan...
For God hath concluded them all in unbelief, that he might have mercy upon all. O the depth of the riches both of the wisdom and knowledge of God! how unsearchable are his judgments, and his ways past finding out! For who hath known the mind of the Lord? or who hath been his counsellor? Or who hath first given to him, and it shall be recompensed unto him again? For of him, and through him, and to him, are all things: to whom be glory for ever. Amen. (Rom 11:32-36)
From here he is addressing the Church as a whole with practical life as a Christian, but not without eventually again proving from Scripture in Chapter 15 that "Jesus Christ was a minister of the circumcision for the truth of God, to confirm the promises made unto the fathers: And that the Gentiles might glorify God for his mercy; as it is written, For this cause I will confess to thee among the Gentiles, and sing unto thy name." (Rom 15:8-9)
Yes, there were Gentiles in the Church in Rome, that is evident from the people Paul greets which he must have met on his travels. Also, Aquila and Priscilla ministered in Rome and certainly would have ministered to Jews and Gentiles having been trained in the Gospel that Paul preached. It does seem however that the Jewish influence and opposition to Gentile converts apart from the Law, demonstrated all throughout the book of Acts, was indicative of the hostile environment in the early church to this ministry of Paul as an Apostle to the Gentile. Paul expected this opposition and wanted to have an opportunity to state his case before he reached Rome.
It was however evident that this opposition never reached the Jewish community in Rome (Act 28:17-21). This may have been absolutely God's intention so that those Jewish Christians in Rome, along with the testimony of Paul's many companions there, would receive this letter with goodwill without the Jewish influences and false testimonies that hindered his Gospel in other places.
Regardless of the actual affections of those in Rome towards him, I thank God that Paul did anticipate opposition because it produced what is probably the greatest defence of the Gospel for me, a Gentiles.
Note for OP's question: From everything I know of the Letter to the Romans and Acts - we cannot discern from it what the social structure of the church consisted of without conjecture. There was almost certainly churches meeting in different homes (Priscilla and Aquilla had a church in their home and there are groupings of people mentioned in his final address), so it is probably as likely as it is today that these churches were fairly homogenous in terms of culture and social classes. Lower classes were more receptive to the Gospel (1 Cor 1:26-28), by God's design, so as a whole the Church in Rome was probably made up primarily of Jews (from various social classes), and Gentiles from lower classes, such as slaves. He does not address "masters" in his letter and uses the word "servant" several times. I am not sure how successful this church was at reaching people from higher social classes before Paul arrived.