I am convinced that a great deal of our confusion comes from the fact that Paul glides seemlessly between a number of different meanings for the word "law" in these passages in Romans and Galatians. Perhaps a good place to start is to remember that the "Law" in general was the Torah, the Pentateuch, the first five books of our Old Testament. When Paul says that the "Law and the Prophets" witness to recent developments (Rom. 3:21), he is referring to the first five books of Scripture, a body of literature.
When we ask what sin was for Paul, we must very quickly get to this Jewish Law. Paul says in Romans 5:13 that sin is not reckoned where there is no law, even though it is present. This comment makes it very clear that one purpose of the Jewish Law was to identify exactly what sin was. Paul confirms this purpose a little later in Romans 7:7 when he says, "I would not have known what sin was except through the law. For I would not have known what coveting really was if the law had not said, 'Do not covet."
A key observation here is that Paul nowhere has some absolute definition of sin. Sin is not for Paul "missing the mark," as if anything short of absolute perfection is sin, any imperfection whatsoever.  Sin is, in one respect, to violate the Law, the Jewish Law, where the Law is a record of the actions that are wrong. As we might expect of the Old Testament, such violations were far more concrete than abstract. They were not some introspective flub where a less than perfect thought flew through your head. We suspect that even when Jesus spoke of looking at another woman with lust (Matt. 5:28), there was a level of intent to act that went well beyond what we would call a lustful thought. 
A great starter definition of the verb, "to sin," is thus "to do wrong," where wrongdoing is defined in the Jewish Law. Closely related is to wrong another person. Thus you can sin against someone (e.g., Matt. 18:21; 1 Cor. 8:12). Such definitions know nothing of the standard "sin every day in word, thought, and deed." This conventional Christian definition of sin does not come from the Bible but from later Christian theology. The Bible consistently treats acts of sin or wrongdoing as something that is avoidable and bad.
To be sure, Paul does not use the word sin in relation to every part of the Jewish Law. Paul never speaks of someone who touches a dead body as sinning nor does he say that someone who works on the Sabbath sins. Indeed, he very likely did not consider such actions to be sins for a believer, especially a Gentile believer.  We thus encounter the key ambiguity in Paul's rhetoric about the Jewish Law: sometimes he used the word "law" in relation to parts of the Law he did not apply to Gentile believers; sometimes he uses the word "law" in relation to parts of the Law that still applied to believers. But he does not clearly distinguish in his language between the two...
 Thus the quite incorrect translation of Romans 3:23 in the New Living Translation where sin is to fall short "of God's glorious standard."
 We have the Romanticism of the 1700s and 1800s to thank for our overly introspective orientations.
 Whether he might have considered them sins for an unbelieving Jew is a question we probably cannot answer given the evidence he has left us.