FWIW, I struggled w/this passage too. I don't speak for my Church on this issue, and I humbly accept and follow their teachings, so take this at face value as little more than my take on your questions.
I agree w/mesue that all teachings in the Bible are valid. But like cookieboo says, we cannot judge what someone's life was before they came to Christ Jesus.
The previous sentences to your Sunday teaching say that a woman is to be silent when around her man, and must be submissive in all things and cannot teach. (1 Tim. 2:11-12). That would preclude even doing the Sunday School teachings. I think the very next sentence blames women for all society's current ills, too (that's my reading of it)
If we take the Bible literally, at every turn, I think we wind up w/a requirement that all priests, ministers, pastors, etc., have to be Jews.
I personally don't have a problem w/women holding high positions in the Church. We have female Lay Extraordinary Ministers (I think that's what they're called; I'm horrible w/names) and no one has any problems w/that. The Church has said in the past that women cannot be priests or even deacons because the laity would be scandalized. I think we're scandalized by other things that happen w/in the Church and in its name more than we would be by having women in high positions. Not only that, if women were put in high positions by the Church by virtue of the Church blessing them w/those positions, it would be accepted by the laity. For me, personally, it comes down to the simple question of whether the particular woman in question knows Christ and the Scripture and is able to spread His word effectively.
Not only that, but the purest followers of Christ were the Blessed Mother and Mary Magdalene. Remember who Jesus first appeared to when He rose on the 3rd day, and how the men didn't believe her. Women are CRUCIAL to the church.
I also have no problems w/divorced people, male or female, holding high office. Unless I know the person, I won't know what led to the divorce. For example, I have to think God does not want people to remain in abusive marriages or those where the children are endangered in any way.
I'm not sure what you meant by background investigations. As mesue said, investigations should be done to ensure no one w/a past that could endanger children should be allowed near children. The only other investigation that should be done is to ensure the person is godly, has Christ in his/her heart, and is effective at spreading His love and word. Again, just my opinion and not that of anyone or anything else.