The Jews respond to Peter’s speech and Peter answers them.
Here the word “they” refers to the people in the crowd to whom Peter spoke.
“when the people heard what Peter had said”
This can be stated in active form. AT: “Peter’s words pierced their hearts” (See: rc://en/ta/man/translate/figs-activepassive)
This means that the people felt guilty and became very sad. AT: “deeply troubled” (See: rc://en/ta/man/translate/figs-idiom)
This can be stated in active form. AT: “allow us to baptize you” (See: rc://en/ta/man/translate/figs-activepassive)
“In the name of” here is a metonymy for “by the authority of” AT: “by the authority of Jesus Christ” (See: rc://en/ta/man/translate/figs-metonymy)
This means either 1) “all people who live far away” or 2) “all people who are far from God.”