What are the connotations of "Vayigash Avraham"?
Rashi: The connotations of "Vayigash Avraham" are threefold - to talk tough, to appease and to Daven. 1
Targum Yonasan: Avraham approached Hashem in order to Daven.
See also the first Ba'al ha'Turim in Vayigash.
What are the connotations of "ha'Af"?
Rashi: It is another word for ha'Gam - "Will You also destroy the Tzadikim as well as the Resha'im?"
Ramban, Targum Onkelos and Targum Yonasan : "Af" - anger. Avraham was asking Hashem whether Hashem in His anger 1 (Onkelos) or Hashem's anger (Yonasan) would destroy the Tzadikim together with the Resha'im.
Moshav Zekenim: Avraham asked why Hashem will destroy the land of the city with the Resha'im. If there are 10 Tzadikim, the land should be saved for them!
Anger here is synonymous with Midas ha'Din (Ramban).
What led Avraham to believe that Hashem would destroy innocent Tzadikim?
Ramban: He thought that Hashem was judging them with Midas ha'Din; he did not realize that He had examined them previously with Midas Rachamim 1 , and found them unworthy. 2 So he suggested that it would be correct to pardon Sedom on account of the fifty Tzadikim that it perhaps contained. 3
Seforno: Because when He said earlier (in Pasuk 20) that 'the cries of S'dom and Amorah are great', it implied that He was punishing the entire town, not just the sinners.
Lev Eliyahu (Bereishis p.59): When there is anger in the world, Midas ha'Din prosecutes Tzadikim, if they do not save their place. Hashem agreed to save the place due to them, i.e. in order to save the Tzadikim. 4
This misunderstanding is evident from the fact that, throughout the current dialogue, whenever Hashem speaks it is spelt 'Yud' 'Key' ... ; whereas whenever Avraham mentions the Name of Hashem, it is spelt 'Alef' 'Daled' ... .
He also asked that initially, Avraham said only that the Tzadikim should live, and he concluded that Hashem should save the entire place for them! His answer answers also this.