Does every human being, at least once in their lifetime, have a genuine chance to repent of their sins, turn away from them and start living a righteous life?
"So having overlooked the times of ignorance, God is now proclaiming to mankind that all people everywhere are to repent" (Acts 17:30). If God expects all people everywhere to repent, then ...
What is the Biblical basis of the idea that men can be given authority to forgive sins?
St. John speaks of it in his Gospel (John 20:19-23):
Jesus Appears to His Disciples
19 On the evening of that first day of the week, when the disciples were together, with the doors locked for fear of the Jewish leaders, Jesus came and stood among them and said, “Peace ...
I was going to make a long comment, but I thought I might as well just give a "short" answer.
I must say firstly, that anyone that claims to have read the Bible and denies Hell's existence, is, at best, deceived, and at worst, insane.
The foundational belief of Christianity is that Jesus died for our punishment, so that we wouldn't have to.
Is repentance of sins possible outside the gospel?
Yes, since God plants conscience in every person's heart. But see next paragraph below.
What does repentance of sins look like for a non-Christian who dies never having the gospel preached to them?
Since they haven't heard a proper presentation of the Gospel, their repentance will be far from what is ...