The slander (gossip) issue definitely the more solid issue in court--boss can fire you; telling everyone violates privacy issues. (Ironically, whoever shared this info probably violated a general moral claus too.)
Now, first off--
"Judge not" is one is one of the most misused statements out there. Sitting there condemning people is out, but the Bible clearly teaches--Judge sin. If a brother sins, and refuses to repent, the church is supposed to discipline them, in hopes that it will bring change. It also tells people to look at teachers/preachers and see if there lives match their words. That is judging. And so on.
"Judge not" probably has more to do with moral superiority. The passage even says, first take care of your own sin, then help your brother to stop sinning. It is hardly "live and let live". The goal is everyone stop sinning and no one is to be morally superior while helping another.
So, in context, if the teacher admits to premarital sex and doesn't repent of it, Bible says the church is supposed to discipline. Now, if she is "asking for forgiveness," (of God not the school), then I can see justification for school having mercy. (Bible even says, that in such cases, the matter can end there.)
2nd, Lot, Abraham, lived before the OT Law was written. So, there was no judgement to kill people involved with extra/pre-marital sex during their lives. It may have still been sin, but no Law punishment was given. (Society had laws, but the Law given to the Jews had yet to be given.) Even if there was, the fact that they had extra-marital sex and lived doesn't invalidate the Bible. It just says that Bible recorded the people breaking more than one Law--the original transgression and failing to follow the Law regarding punishment.