Not all wounds. Just like you will never grow back your arm if it's cut off, if significant damage is done to you psychologically during your early development then that skews the rest of your adult life. It can't be undone, you didn't develop properly. You can grow and learn to make that wound a smaller and smaller part of who you are, but it will always be there.

As for trauma endured in adulthood, I don't know. I think it's really quite similar. Most people never actually move on after losing a loved one, for instance, they just learn to find a new normal and try to work through their mourning in a healthy way. I think even if you were granted immortality, these events would continue bothering you hundreds of years later. You might just think about them less frequently, or eventually forget, but the pain would still be there.