This chapter reads a bit like "friendly marital advice from bachelor Uncle Paul." There are some unspoken and barely spoken assumptions that are behind Paul's teaching in this chapter. He believes that the "Day of the Lord" is soon to come and he understands that day, not as joy and peace, but death and destruction. At the end, when Jesus returns, all will be restored, but it will be hard times and difficult experiences until then. Remember that by the middle of the first century (AD 60 or so) Rome has turned decidedly against the fledgling Christian movement. Nero has blamed the Christians and killed them by the thousands. Times are already difficult. In light of this situation, Paul is suggesting that the Corinthians not take on any more responsibility and to keep their lives unencumbered as much as possible.
7:14 is one of the underpinnings of the theology of infant baptism. If the faith of the believing parents makes the children holy than these children should be baptized and included in the church. It is an interesting argument and, fortunately, not the only argument.
Notice that even though Paul suggests that we "stay in the situation/station" we were in when we became Christians, he allows exceptions in almost every case. To the slaves, stay as slaves (but get free if you can) is one perfect example (21ff). This is part of the "trajectory against human slavery" that emerges by the end of the first century.
Notice here the second grounds for legitimate divorce. The first is adultery. Here in chapter 7:15 we add abandonment: he the unbelievers leaves, let it be so. The brother or sister is not bound in such circumstances . . . " For the believing couple (where both are believers) there are a different set of understandings but for the "unequally yoked" marriage abandonment is grounds for divorce and remarriage by the believer.