Seeing as they had just appointed Lot a judge (Rashi, Pasuk 1), how could they now accuse him of judging them?
The judges of Sedom were not conventional judges, who sought the truth and acted upon it. They judged by the evil, cruel laws of S'dom. In fact, Lot's defense of his guests clashed head-on with the pervert laws of S'dom, which were designed to strip the visitor of his money, his dignity and even his life, if it suited them. 1
As Eliezer Eved Avraham discovered when he once visited S'dom, as related in the Midrash.
Seeing as the word "Vayiftzeru" has connotations of appeasement, what were the people asking Lot here?
Ramban: Either to open the door to let them in, or to move aside and give them access to enter.
Why does it say "they said... they said..."?