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 needed for Christian conversion. They may simply repent from being lazy, or even as trivial as repenting from a habit of driving over the speed limit. I did both when I was young, but those repentance didn't save me because that's NOT the kind of repentance needed for salvation. Even a self-serving atheist can repent from that kind of "sin".
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? If so, does this also include people who never hear the gospel?
Again, without proper gospel presentation, it depends on what "righteous life" means. Every religion has their own version. It can simply mean living a decent life following the ethical standard that either myself or my preferred society defines for me. That is NOT what righteous life means to Christianity, which is living according to God's standard, not my own standard. Genuine chance requires genuine understanding of the magnitude & character of the offense, which in Christianity means:
- realizing that we have been following our human made standard, even though it can be very decent and sacrificial
- realizing that the main aggrieved party is God our creator, not only our fellow human beings
- realizing that the main offense is our failure to center our whole life to honor and love God instead of ourselves, not merely the failure to follow some moral code, even if the moral code is already one of the best (such as the OT code)
So the answer is: Yes, to those who understood properly the gospel message, No to everyone else.
Proper Christian repentance is intimately linked to faith in Jesus Christ (see this article). But the object of our faith (Jesus Christ) needs to be communicated FIRST by the gospel presentation, which in turn genuinely informs the challenge for proper repentance that leads to faith, which in turn leads to salvation. So this question is masking an underlying question: Will everyone who has not heard the gospel before they die have a genuine chance to hear it? which has been answered in your other question: What is an overview of Christian viewpoints on the eternal destiny of individuals who die never hearing the Gospel?