There isn't a 'Vermint city' in soulsilver. I think you mean Vermilion city. Well, there is a gym, and that is where you board the SS.Aqua.
no such thing not in soul silver
Saffron City
In Cinnawood City
at olivine city
swim
olivine city
In soul silver and heart gold you find it in euritik city in one of the houses
you do ecruteack city
It is in golden rod city
Cerulean City
in cianwood city
South.