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 Sedom. In fact, Lot's defense of his guests clashed head-on with the pervert laws of Sedom, 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 visited Sedom, 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..."?
Moshav Zekenim: They said "Gesh Halah" to words that they were pleased to hear (do to my daughters as you wish), and they said "one came to live - will he judge...?!" to words that they did not like hearing (do not harm these men). 1
Hadar Zekenim and Riva say similarly.
רש"י: על הבנות, אמרו לו גש הלאה...ועל שהיה מליץ על האורחים אמרו האחד בא לגור: מנין לחלק דבריהם- "גש הלאה" על הבנות, ו"האחד בא לגור" על שהליץ על האורחים?
גור אריה: "גש הלאה" משמעו 'לך מעִמדי', ואין לומר כך כאשר רוצים לדבר עמו ולומר לו "האחד בא לגור", ולכן פירש רש"י שדבריהם מתחלקים- כלפי הבנות וכלפי האורחים.