Do a little figuring about what we're told of Lot and his family, and you'll notice that Abraham had a good mathematical reason to stop his haggling with God over Sodom's fate upon securing a promise not to destroy it if even ten righteous people could be found there. Counting Lot and his wife, his two unmarried daughters and their fiances, and at least two other married sons or daughters (as suggested in Genesis 19:12) and their spouses, we have at least five couples in Lot's immediate family, which comes to ten people. Had they all been at least as righteous as he was, instead of dismissing Lot's warnings of imminent doom as some kind of joke, God might have spared Sodom on their behalf.
What did Lot and his daughters have to eat and drink after all surrounding civilizations but Zoar were destroyed? Well, the civilizations were destroyed, but not necessarily all of the countryside around them; some of the animal herds likely survived and were roaming loose without their former owners around to round them up, so Lot and the people of Zoar likely could begin gathering free herds to themselves. As for the drink, some of the animals would be giving milk, and volcanic ash from the destruction of the cities would have fertilized the land and made it a great place for growing fruit... such as wine grape vines, for one.
They also had access to a supply of salt...
After Joseph is accused by Potiphar's wife of raping her, a very angry Potiphar has Joseph imprisoned. While this is still unjust to Joseph, consider that in ancient societies simply being accused of sexual assault would usually result in death for the slave involved. Though the text doesn't explicitly say this, Potiphar evidently realized that his wife was lying and spared Joseph's life as best he could.
How do we know anything about what happened with Lot and his daughters, considering that they never went back to visit his uncle Abraham's family? Probably because, like many other peoples, the Ammonites and Moabites regularly told their origin story to visitors themselves. Other people such as the Greeks weren't too shy to tell stories about incestuous relations between various famous ancestors and gods and rulers of theirs, so the Ammonites and Moabites probably weren't too ashamed to tell their tales to the Egyptians and Hebrews about the strange way their nations got started either.
The men of Sodom habitually gang-raped any strangers who passed through their town. How many unsuspecting travelers and visitors did they molest before they were dumb enough to try it on some angels?
Enough that God is convinced that Sodom must be completely and utterly obliterated from the face of the Earth. So probably a lot.
Why do we get no record of what Lot had to say about his daughters turning up pregnant when he knew there was nobody else around to impregnate them? Possibly because he wasn't around anymore by the time their pregnancies began to show. They did say he was already getting old.
He might have had very little, or nothing at all, to say about it in the first place. Both daughters had recently had fiancés, so Lot may have assumed that turning up pregnant soon after revealed that they had not been virgins at the time of the destruction after all, and downplayed the situation in an effort to preserve their honor. Alternatively, supposing that he was aware of the truth about how they got pregnant, it's not uncommon for rape victims to keep secrets out of embarrassment.
After slaughtering all the Shechemite men in revenge for Dinah's violation, Simeon and Levi carried off the surviving women and children, probably as slaves. That's a rather terrible thing to do to those people in addition to the first crime of slaughtering so many men who'd had nothing to do with violating Dinah, but consider what Simeon and Levi's options were once all the men of Shechem were dead:
They could simply leave the women and children there. When word got out that the women and children of Shechem no longer had any men to defend their city, every roaming band of raiders in range would surely have descended on the city to seize the survivors for themselves, probably for a massive rape-fest and possibly followed by their murdering everyone.
They could simply slaughter all the women and children along with the men, something that not even their vengeful wrath over a single man's violation of their sister could really justify to them.
They could take the surviving women and children with them, cumbersome and troublesome to the family as that might be. This is what Simeon and Levi ultimately decided to do; and arguably the best of the three options, though still a rather bad one.
