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 (in spite of the dangers involved should the men of S'dom discover what he was doing).
Rashi #2: He was warning them to take precautions against the Sedomians finding out that they were there.
Ramban and Targum Yonasan: He was asking or even pleading (Ramban) with them to be his guests.
Why did Lot ask them to wash their feet only after having stayed overnight, particularly when a little earlier, Avraham had reversed the order?
Rashi #1: 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 #2 (on Bereishis 18:4): Mistaking the angels for 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 Avraham tell the angels to leave first thing in the morning?
Ramban #1: Lot foresaw what the men of S'dom would 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 S'dom, so he informed them that they were welcome to leave first thing in the morning, should they so wish.
Considering that they immediately accepted Avraham's invitation, why did they initially decline that of Lot?
Rashi and Ramban #1 (both citing the Midrash): The Midrash learns from here that 'One refuses the offer of a 'Katan' 1 but not that of an Adam Gadol'.
Ramban #2: Knowing that Lot would persist, the angels initially refused, in order to gain him more reward, for his amazing display of Hachnasas Orchim.
Rashi (in Pesachim, 86b): And they only accepted his invitation after he pleaded with them a long time, as the following Pasuk testifies.