Romans 2:12 For all who have sinned without the Law will also perish without the Law, and all who have sinned under the Law will be judged by the Law;
Romans 5:13 for until the Law sin was in the world, but sin is not imputed when there is no law.
So, is Paul saying that sin is a "byproduct of the Law" or is Paul saying that until the Law sin was in the world, but not imputed as a transgression of the Law?
Continue the passage you started -
Romans 7:10-11 I was once alive apart from the Law; but when the commandment came, sin became alive and I died;
10and this commandment, which was to result in life, proved to result in death for me;
11for sin, taking an opportunity through the commandment, deceived me and through it killed me.
Is Paul saying that the Law produced sin or that sin took opportunity through the commandment?
Paul's point is that sin existed from Adam forward, but those apart from the Law did not sin as Adam sinned (their sin was not a transgression of a commandment, but it was still sin).
The Law magnified sin. Sin even increased under the Law as things became sin for those under that covenant.