The chapter
Simply Paul argues a case about Abraham. Genesis says Abraham believed God and it was counted to him as righteousness; Paul asks when that was said, and answers that it was before circumcision and before the law, so it cannot have been a reward for either.
Context The chapter is an argument, not a story. It follows the claim at the end of chapter 3 that a person is justified by faith apart from works of the law, and it works by reading one sentence of Genesis very closely.
Tradition This chapter is where the churches divide, and each states its position in its own documents. Reformed churches confess that God justifies by pardoning sin and accounting a person righteous, not by infusing righteousness into them, and that faith is the alone instrument of that justification, though never a faith that stays alone. The Church of England says the same in fewer words: we are accounted righteous only for Christ's merit, by faith. The Council of Trent reads Paul's own phrase differently, holding that when the Apostle says a person is justified by faith and freely, the words must be understood as the Catholic Church has always understood them, and its canons condemn the position that faith alone suffices with nothing else required. Philaret's catechism, the Orthodox witness ingested here, does not frame the question this way at all: it speaks of those who do not count themselves righteous nor rest on their own good works, but hunger for the justification of grace through Jesus Christ. That is a different register, not a fourth answer to the Western argument, and it is left as it stands rather than pressed into the shape of one.
Verse 3
Simply The sentence Paul builds everything on is a quotation from Genesis: Abraham believed God, and it was accounted to him for righteousness.
Context In Genesis the sentence sits between a promise of descendants and a covenant ceremony, and nothing in the chapter marks it as a turning point. Paul makes it one.
Verses 9-12
Simply The argument turns on timing. Genesis records the sentence about Abraham's faith before it records his circumcision, so Paul concludes that the blessing was not attached to the sign.
Context Circumcision is commanded in Genesis 17, two chapters after the sentence Paul is quoting. The order of the two chapters is the whole of the argument.
Verses 18-22
Simply Paul reads Abraham's faith as trust held against the evidence: he was old, the promise was of descendants, and he kept believing it.
Context Genesis is less tidy than the summary. In it Abraham laughs at the promise of a son, and Sarah laughs too. Paul is reading the outcome of the story rather than every scene in it.