Are there any verses in the bible that would support/condemn a woman speaking in a Bible class or Bible study in the church (not worship services) with a Christian man present?
I often wonder if we can separate Bible classes (as in "Sunday school classes") from the assembly of the church ("the worship"). It seems to me that any time the church gathers together, even if separated into age group classes, then Paul's words about it being a shame for a woman "to speak in the church." (1 Cor 14:34-35) However, that is my opinion, and I am not dogmatic even about that.
In most congregations where I have worshipped the idea was that if we err, we do so on the side of caution. Therefore, women were allowed to teach the younger age groups in a Bible class, but men generally taught the classes above the fourth grade (ages 8 or 9) because it was likely that some of the male students would have been baptized about that age; if there were Christian "men" even of that age in the class, women should not teach.
The scriptures are not clear on what constitutes keeping silent in the church. Most congregations I have attended have preferred to be cautious in interpretation, rather than liberal.
Individual Bible studies, on the other hand, outside of the context of the assembly of the church, are a different matter. It seems fairly clear that Christian women were allowed, along with Christian men, to teach in a private setting. In the first century, as now, that is where most conversions happen. The scripture that would apply here is Acts 18:24-26: "And a certain Jew named Apollos, born at Alexandria, an eloquent man, and mighty in the scriptures, came to Ephesus. This man was instructed in the way of the Lord; and being fervent in the spirit, he spake and taught diligently the things of the Lord, knowing only the baptism of John. And he began to speak boldly in the synagogue: whom when Aquila and Priscilla had heard, they took him unto them, and expounded unto him the way of God more perfectly." Both the husband and wife are here shown to be, at the same time, involved in teaching another, privately.