I feel like Randall has known he was different from the beginning. I mean, being a different race than everybody else in the family could make you prone to pick up on things that would make you fit in better. So early on, he decided that role was as the dependable, stoic, protector (as he was with Rebecca after Jack died). Yes, he was probably hard wired to some degree to take on that personality, but his situation brought it out more. His role was to be the perfect kid that didn't cause trouble.
After he beat up the mugger, he probably realized he was strung a little too tight and that, no, running was not going to take care of his anxiety. He was uncomfortable being called a hero because he knew he sort of lost it with that guy -- cutting loose with a lot of built up tension he had fooled himself into thinking he was dealing with. That fight showed him he was wrong and he didn't like the attention it brought.
Im confused about how the robber got that jewelry. IDK if he had broken in there before and got them then, and was coming back or if he came back later after Randall confronted him. It seems unlikely to me that he could have done it the night Randall confronted him. Plus, they made a lot of noise about how robbers will often hit the same place twice.