Does Islam support the killing of disbelievers when they hurt your feelings? Is Islam truly a religion of peace?
I think the question needs more explanation and details. What do you mean by "hurt your feelings"?
In general, you cannot kill a non-Muslim or Atheist or any human being unless there is a justified cause according to Islamic law. This is because Allah (the Most High) has commanded in Qur'an:
In the tafsir of this Ayah, Ibn Kathir (Rahimahullah) mentions several hadith where the Prophet (sal'allahu alayhi wassalam) prohibited, warned and threatened against killing a Mu`ahid, i.e., non-Muslims who have a treaty of peace with Muslims. One such hadith is:
In another hadith, he (sal'allahu alayhi wassalam) says :
Now, you haven't mentioned in your question if the "non-muslim" has a treaty or agreement with Muslims or not. You haven't even mentioned what crime the non-muslim has done. "Hurting the feelings" could mean a lot of things. My feelings could get hurt if my boss scolds me. That is not a crime.
Islam is not what you think or I think or anyone thinks. Islam is what came through revelation and what was instructed by the Prophet (sal'allahu alayhi wassalam). And yes, Islam is a place for peace.