Who were the three men who came to visit Avraham?
Rashi, Ramban, Rashbam and Targum Yonasan: They were three angels. 1
Micha'el, Gavriel and Refael. See Ba'al ha'Turim.
Why was it necessary to send three angels? Why would one not have sufficed?
Rashi, Ramban and Targum Yonasan: Because they came to perform three different missions, 1 and one angel cannot perform more than one mission at a time. 2 Consequently, whenever the Pasuk refers to the actual Sh'lichus, the Pasuk switches to the singular.
One to inform Sarah (or Avraham - Targum Yonasan) that she would give birth to a son one year later (Micha'el - See Ba'al ha'Turim), one to overturn S'dom (Gavriel) and to heal Avraham and to rescue Lot (Refa'el). Riva says that healing and rescuing fall under the same category (as is evident in the B'rachah of 'Refa'einu' in the Amidah]). He asks why Avraham himself did not tell Sarah. He answers that he was afraid that she would laugh.
Why did Hashem send them in the form of human-beings?
Rashi: To fulfill Avraham's desire to feed guests
Why does the Torah write the word "va'Yar" twice?
Why did Avraham prostrate himself before his guests?
Seforno: On account of their awesome appearance - In fact, he assumed them to be emissaries of some important king or other.
Chochmah u'Musar (2 p.404): Even though he was a king, he lowered himself 1 and called himself "your slave" (Pasuk 5), due to his great desire to go in Hashem's ways to bestow Chesed - they will understand, all the more so how much Hashem Himself wants to bestow Chesed!.
If he was a kin, how could he do that in light of the Gemara in Kesuvos, 17a 'Melech she'Machal al Kevodo Ein Kevodo Machul!'
Regarding Avraham, the Pasuk calls them 'men'. Why regarding Lot, does it call them 'angels'?
Rashi #1 (on Bereishis 19:1): When the Shechinah was with Avraham it calls them 'men': when it was not with Lot, it calls them 'angels'.
Rashi #2 (Bereishis 19:1): Specifically in the presence of Avraham, who, due to his greatness, was accustomed to seeing angels no less than men, it calls them 'men'.