What did Lot mean with his strange introductory comment "Hinei Na Adonai!"?
Rashi #1: Following in the footsteps of his uncle Avraham, he was informing the angels that, having reached his abode, he was at their service 1 (in spite of the dangers involved should the men of Sedom discover what he was doing).
Rashi #2: He was warning them to take precautions against the Sedomians finding out that they were there.
According to the Ramban and Targum Yonasan, he was actually pleading with them to be his guests.
Why did Lot suggest that they wash their feet only after having stayed overnight? Avraham suggest that they wash their feet immediately!
Rashi: To protect himself. So that, should the Sedomians arrive and see that they had not yet washed their feet, he would be able to claim that that they had only just arrived, thereby minimizing their anger.
Rashi (on Bereishis 18:4): Mistaking the angels as Arabs, who bow down to the dust of their feet, he asked them to wash their feet prior to entering, because he was particular about allowing Avodah-Zarah into his house, whereas Lot was not fussy.
Hadar Zekenim (on Bereishis 18:4): Lot wanted them to enter his house as soon as possible, due to danger if people of Sedom will see them.
Why did he tell the angels to leave first thing in the morning?
Ramban #1: Lot foresaw what the men of Sedom would likely do to them, only he assumed that they would arrive only in the morning.
Ramban #2: He saw them as travelers, who, he assumed, did not want to sojourn in Sedom, so he informed them that they were welcome to leave should they so wish.
Considering that they immediately accepted Avraham's invitation, why did they initially decline that of Lot?
Rashi: The Midrash learns from here that 'One refuses the offer of a 'Katan' but not that of an Adam Gadol'.
Ramban: Knowing that Lot would persist, the angels initially refused, in order to gain him more reward, for his amazing display of Hachnasas Orchim. 1
The Ramban also cites the Midrash cited by Rashi.
רש"י: הנה נא אתם אדונים לי אחר שעברתם עלי: למה פירש רש"י כך?
גור אריה: "נא" פרושו 'עתה', ולכן פירש 'הנה אתם אדונים לי עתה מאחר שעברתם עלי'.
רש"י: הנה נא צריכים אתם לתת לב על הרשעים הללו שלא יכירו בכם: מה כוונת רש"י?