First bit - yes, pretty much what I'd assume happened, basically dumbassery. I'm not sure that the squad have rounded on Sterling though, I don't see anything supporting that. Henderson has come out of it okay, and deserves some credit if that's true - he's probably the squad member who knows each best.
London? No idea, I didn't know Gomez was from there!
It may well have been different for a big game, but we'll never know until that happens. I think a manager making a stand on discipline isn't a bad thing, and he'll have to do similarly in the future to not be completely exposed; again, only time will tell. I don't think Southgate cares too much about how it was viewed externally, but instead is pretty annoyed that it leaked, forcing the acknowledgement and media storm. Worse that storm is built on leaks, which may be partisan and partial in their presentation.