Honestly, what on earth do lawsuits have to do with Sexual Immorality? Why would Paul talk about immorality and the lack of church discipline in Chapter 5 and then go back to immorality and what we need to do with it in Chapter 6? Unless of course, Chapter 6:1-8 connects to Chapter 5. That it somehow is a part of a larger theme. It seems what Paul is communicating is that we represent something bigger than ourselves; something not about us but about God; something of profound significance. That the way we live reflects Him. The way we interact reflects Him.
If we find ourselves tattling to our parents, proverbially speaking, when we can’t get along together, it dishonors God because we’re proving that the Gospel has no life change in us and the Gospel doesn’t affect the way we live. Therefore, we need (in context of 1 Cor 6) the pagan courts to determine who is right and who is wrong instead of figuring out by the grace of God in us how to live together in unity.