Nothing is absolute. Most bullies arent unrepentant, evil little ****s. They have psychological/emotional issues that cause them to act out. And yes, things CAN cause them to snap out of it, or see that someone is hurting and offer condolences. They may act like jerks, but theyre still human.
In high school, there was a guy who was just a total dick to me. I wouldnt call him a bully persay, he just didnt like me for some reason and made sure I knew it on a regular basis. When a family member unexpectedly died of cancer, he laid off and eventually even apologized. We didnt end up as friends but we never had a problems after that. But wait, thats totally unrealistic, so it must have all been in my head. 
 
After Peter humiliated Flash, Flash saw that Peter was on equal footing  he wasnt that shy, weird and defenseless kid Flash thought he was. He gained respect for Peter, which allowed him to be more open to seeing Peters pain. Remember, this happened in the comics - Flash did become Peter's friend, and in the film, all Flash did was try to apologize (and it sounded like he was even going to sympathize with Peter's grief, which makes it even more understandible).