What did Hashem mean when He said that He 'knew' Avraham'?
Rashi and Moshav Zekenim: He meant that He endeared, or befriended, him. 1 Nevertheless, it retains its simple meaning of 'knew him' because when a person loves his friend he familiarizes himself with him and brings him close.
Ramban: It means literally that Hashem knew Avraham - because in the realm of Hashem, knowing is synonymous with Hashgachah P'ratis (personal supervision). 2
Targum Onkelos 3 and Targum Yonasan: He meant that Avraham Tzidkus was known, or revealed, to Him, as the Pasuk goes on to explain.
Hashem constantly watches over Tzadikim personally, as the Pasuk says in Tehilim (33:18) "Behold the eyes of Hashem are on those who fear Him"). Indeed, here he says that others are subject to Mikrim - but you cannot say that they are left to nature - the Ramban at the end of Bo says that everything is Nisim, even for a Yachid, without Teva - if he sins, evil will pursue him...
This is according to our text in Targum Onkelos. Rashi asks that according to this, the word "Lema'an" is meaningless.
What is the connection between the first two words and the rest of the Pasuk?
Rashi and Seforno: Hashem chose and loved Avraham and divulged what He was about to do to S'dom, because as a result, he would teach his children to do Tzedakah and Mishpat, 1 just as he himself was about to display. 2
To whom are "Banav" and "Beiso" referring?
Rashi in Sanhedrin, 57b: "Banav" is referring to Yitzchak; "Beiso", either to Ya'akov or to the women in his family. 1
Targum Onkelos and Targum Yonasan: "Banav" is referring to his children; "Beiso", to members of his household (his slaves and maidservants).
Sarah, Hagar and Bakol.
Why does the Torah say 'concerning Avraham' (rather than 'concerning Avraham's family) even though it is referring to an era when he will no longer be alive?
Rashi: We learn from here that if a deceased father leaves behind a righteous son, it is as if he is still alive.