"behind him" or "after he went through"
I beg you, my brothers
"I plead with you, my brothers"
Lot spoke in a friendly way to the men of the city hoping that they would listen to him. AT: "my friends" (See: rc://en/ta/man/translate/figs-idiom)
do not act so wickedly
"do not do something so wicked" or "do not do such a wicked thing"
"Pay attention" or "Look here"
have not slept with
"have not engaged in sex with." Your language may have a more polite way of saying this. AT: "have not known" (See: rc://en/ta/man/translate/figs-euphemism)
whatever is good in your eyes
"whatever you desire" or "whatever you think is right"
under the shadow of my roof
The two men were guests in Lot's home, so he needed to protect them. The word "roof" is a synecdoche for the whole house and a metaphor for Lot protecting them. AT: "into my house, and God expects me to protect them" (See: rc://en/ta/man/translate/figs-synecdoche and rc://en/ta/man/translate/figs-metaphor)