It is important to note that chapters were added to the biblical text by later copyists and editors for ease of navigation and citation, not by the original authors themselves. The chapter break between 1 and 2 is artificial, for 2 continues the train of thought found in 1.
As Paul develops his narrative that mankind has descended into sin, he establishes that sin is universal. All humans are tainted by it, both Jew and Gentile.
In establishing that sin is universal, Paul can then develop this point to show that all humans need salvation if they are to survive the judgment of God. This idea Paul then focusses on Israel to expose hypocrisy.
He attacks the idea that one does not need salvation if one adheres to the Law of Moses, the Torah. It is important to understand the nuance here. Paul is not saying that the Torah is of no value. Nor is he saying that a person who keeps the Torah perfectly need fear the judgment of God.
Rather, he is saying that one must keep the Torah perfectly in order to withstand the judgment. But since no one can keep the Torah perfectly, as is evident from the sins of Israelites, one cannot be saved by attempting to keep the Torah.
This will become clearer in subsequent chapters, where Paul will discuss justification in more depth. At this stage, he is establishing that (1) Israelites are in need of salvation, just like the Gentiles, and (2) Israelites should be humble before God because of point (1), especially in light of the fact that the Gentiles themselves obey the moral precepts of the Torah (albeit imperfectly).
Paul is writing to a mixed audience, and he is trying to build a community where there is neither Jew nor Gentile in Christ, but all love one another as Christ loved us. (Although, as we shall see later, Paul’s doctrine of Israel is complex.)
Paul reminds his readers of the eschatological consequences of sin, for those who do not receive salvation: judgment during the Day of the LORD and in the hereafter. This note of warning and sobriety is another way of impressing upon the audience its need for salvation.
All Along the Watchtower 
The division here moving from Chapter 1 to Chapter 2 is quite reasonable.
In 1 v 18-32, could be loosely described as `the depraved man’, the `Don Juan’ type figure, whose behaviour has degenerated into that which is utterly disgusting. Paul refers to these people in the third person. They are in opposition to God and they know it.
In 2v1 – 16 there is a shift and the topic could be loosely termed `the good pagan’. There is a shift, where Paul addresses these people in the second person, `you’, as if he is responding to a heckler in the crowd. These are people who are trying to do good and who think, that by virtue of this, they can work their way into the heavenly kingdom.
2v17 – 29 moves on to the religious person – and again he shows that you don’t see life through religion; those who believe that they will see the heavenly kingdom as a result of their religious observance stand condemned.
You’re right that the whole section Romans 1v18 – 2v29 shows that everybody in and of themselves stands condemned; perhaps the chapter division might have been better after 1v17, where he has finished his statement of the theme of the gospel
I am not ashamed of the gospel, because it is the power of God for the salvation of everybody who believes; first for the Jew and then for the Gentile. For in the gospel a righteousness is revealed, a righteousness that is by faith from first to last, just as it is written: “The righteous will live by faith”
The division at the end of chapter 1 and beginning of chapter 2 seems reasonable to me, though.
LikeLike