Alfred and James's old hideout represents their past solidarity and brotherhood. It also represents a time when they were younger, when they were just kids whose lives were not nearly as complicated as they are now. The hideout is a throwback to childhood and to feeling safe and secure. Alfred finds James there when James is hurt and hiding from the police, as if James had tried to retreat into his childhood and thus to a time when he was safe. James had nowhere else to go when he was alone and scared, and so he traveled back in time to the last place he remembered feeling secure.
The robe symbolizes Alfred's legitimacy as a boxer. When one does not deserve to be in the ring, one does not get to wear a robe with his name on it. Alfred is very moved when he earns a robe because it gives him an identity he has never had. He is also moved by the fact that Henry bought it for him, acknowledging the fact that his efforts are being noticed by others.
The three steep flights of stairs to Donatelli's gym represent a long and uphill struggle to get somewhere. Alfred's journey throughout the book is much like those stairs—difficult to climb, easy to fall down, and sometimes seeming like the top is too far away. However, just as climbing those stairs into Donatelli's gym yielded positive and unexpected results, pushing himself to keep going finally allows Alfred to find that for which he is looking.