Not always. For example, say I was a victim of child rape (I wasn't, but pretend I was for the purposes of this discussion). I think I would have every right to be harsh with the person who assaulted me, and make sure they were punished within the full extent of the law. Likewise, I don't think forgiveness for someone who would do that is necessarily a good thing. I doubt very heavily that if I was in that position that I'd ever forgive, and I'd be ok with that.
That's an extreme example, I could also completely sympathize with a person who would never forgive a cheating spouse, or a negligent parent. Blanket forgiveness is not a good thing, however it would still be the right thing to do for the wrongdoers to own up to their mistakes and at least attempt to make things right.
As for your second part, I disagree we are all ultimately sinners before god, and we have wronged god... that kind of comes with the atheist territory

That being said, I view that to be a very sad and depraved worldview. Most people are good people who try to live the best life they can. Are we perfect? No. Are we generally good? Yes.
I think attempting to make amends is the only morally permissible action when you have wronged someone. Failure to do so is to abdicate responsibility for your actions.