If you were a rabbi in Judea in the 1st century B.C. and I asked you how I might obtain eternal life, what scripture would you show me? In other words, what was the difference between a "believing" Jew attempting to follow Mosaic Law and a "non-believing" Jew attempting to follow Mosaic law and what was the Old Testament scripture which differentiated the two?

The difference between a believing Jew and an unbelieving Jew (assuming we are not dealing with idolatry which, in the first century B.C., was no longer a Jewish problem) would be recognizing the actual purpose of the Mosaic Law. The true believing Jew would put his faith in the God of Israel and trust the God of Israel for his salvation in the same way Abraham did in Genesis 15:6 where we are told that Abraham believed God and it was reckoned to him for righteousness. After the believing Jew put his faith in the God of Israel and trusts Him alone for his salvation, he would then see the Law as being a rule of life for one who is already saved, rather than a means of salvation. The unbelieving Jew, however, would view the Law as a means of salvation and would, therefore, put his trust in his own works. That is why Paul spent so much time in Romans distinguishing between salvation by works of the Law and salvation by grace through faith. The unbelieving Jew has put his faith in his own works earning salvation, whereas a true believer would recognize that he can do nothing to commend himself before God and, therefore, realizing God’s mercy.

Wasn’t capital punishment prohibited when the Jewish leaders stoned Stephen at the time of Messiah?

The stoning of Stephen took place at an “opportune” time, when a Roman procurator had passed away, and the new procurator had not yet arrived on the scene. With no procurator to control matters, the Sanhedrin – through a mob scene – were able to carry out Stephen’s execution, though it was quite illegal.