What entered the heads of Lot's daughters to commit incest with their father?
Rashi and Rashbam: They believed that (a) the entire world had been destroyed 1 and that they were the only three survivors, 2 and (b) that Lot was already an old man, and that they need to act immediately before Lot would be unable to father children. 3
Ramban: They believed that HaSh-m had saved them for that purpose, and they had faith that HaSh-m would crown their efforts with success and that they would give birth to a son and a daughter, who would marry and begin the world afresh, (in the same way as Kayin and Hevel had been allowed to marry their twin sisters) - based on the Pasuk "Olam Chesed Yibaneh."
Da'as Zekenim citing Ibn Ezra: They considered themselves to have good lineage, and thought that there was nobody proper for them to marry. 4 Also, people would not want to marry someone saved from the destruction. They were not worried about perpetuating the world, rather, their family.
With a flood of fire.
In light of what the angel had said to their father (to escape to the mountain because that was where Avraham lived) however, how could they not have known that Avraham and his family were still alive?
Note also, that, strictly speaking, a ben Noach is permitted to marry his daughter (although it is looked down upon as immoral [Refer to 20:12:2:1]). Chochmah u'Musar (Vol. 2, p. 339) - because later the Torah would forbid it, it spiritually marred Amon and Mo'av to the point that (even after conversion) they cannot marry into Yisrael. Refer to 36:5:1:1*. Even a light sin does so much!
Seforno: They thought that, due to their father's advanced age, he would not have the strength to move to another area, and in the area where they currently lived, there was nobody who was fitting for them to marry. It was customary to marry someone of a similar status to oneself. He explains (to 19: 37-38) that they called their sons 'Mo'av' and 'Ben Ami,' to indicate that they did not become pregnant from someone who was below their dignity to marry.
Why did they not consult with their father?
Ramban: They did not want to do that due to their modest disposition, since, even though having relations with him was not prohibited, it was very much looked down upon and was simply not done.
Why does the Pasuk refer to the older sister as the firstborn (Bechirah), seeing as she had two married sisters in Sedom, who presumably, were older than her?
Ramban (citing the Ibn Ezra): It seems that he married Iris (the mother of the two sisters who were saved), after his first wife (the mother of his two married daughters) died.
Ramban and Targum Onkelus: It is customary to call an older child "Bechor," 1 as opposed to the younger one, even if he (she) is not actually the firstborn. 2
Hence both Targum Onkelus and Targum Yonasan translate "ha'Bechirah" here as 'the older one.'
See for example Yehoshua 6:26).
