I always thought of a person’s “old stamping ground” as being a larger area that a person used to frequent. For example, the downtown area of my city is my old stamping ground. It’s quite large, but that’s where I used to spend most of my time. I went to university there, I went to parties there, I had my studio there, I practiced martial arts there, a frequented museums there, but these were all blocks away from each other.
I think of a hangout as being one small place, like a café, a bar or even a small park, where a lot of people come to loiter.
In the US we both stamp and stomp. There seems to be a subtle difference between the two, but it’s too early in the morning for me to think of what it might be. I think stomping is more injurious to something or someone than stamping is, but not always.