I'll limit this to... forgiving an act less serious than murder or bodily harm, and much more serious than simple name-calling.
2/. How forgiving are you?
I can easily forgive and forget someone who hurts me personally, partly because I may have allowed it to happen.
I assume a person is a good person when I first meet them, and take them at their word. It's worth the chance of having one hurtful experience, if in exchange you can find out quickly if someone is dishonest, or out to take advantage. Much better than spending your life being defensive and suspicious of everyone. Very seldom have I been hurt. If I was, I moved on quickly from that person, and forgot it completely. I can't even remember a case right off hand, for that very reason. I learned this from a friend.
It's more difficult for me to forgive someone who hurts another person. But that's just instinct or human nature, with most people, I think. I can forgive that fairly easily too... as time passes...if it doesn't happen again.
.. If it continues, or expands to hurting even more people, then it becomes #3/. below...
3/. Is there such a thing as being too forgiving?
In my opinion, Yes. There are people who are so emotionally messed up they can disrupt the lives of everyone they come into contact with, their entire lives. They usually target the nicest most vulnerable people, because they have learned how to spot them quickly. Frankly, I would feel like a traitor and a hypocrite, if I forgave people like that. Although it's true that emotions involved in not forgiving someone can be debilitating if you dwell on them, I don't dwell on them. Not liking yourself is much more debilitating and constant.
I should add, that I've only known maybe 3 or 4 people like that my whole life. None of them was ever able to change, or wanted to.