Does God forgive a murderer who wrongfully takes a life? Is that considered justice for the victim and their family, and what is the standard of justice with God?

The right of the murdered person cannot be lost with Allah Almighty. The first matter to be judged among people on the Day of Resurrection is the sin of murder. Sincere repentance from the killer erases the right of Allah Almighty, but the right of the murdered person is preserved and does not fall. He will receive his due on the Day of Resurrection, either by taking good deeds from the killer, or by the killer bearing the murdered person's sins, or by Allah compensating the murdered person from His bounty with what satisfies him. This confirms the perfection of Allah's justice, where no one is wronged even an atom's weight.