I think the Master is a renegade Timelord. He probably has a TARDIS tucked away on the island somewhere, let's see fi we can play 'spot the blue police call box.'
Kudos to the people who called Punk Hazard as the site of the Aokiji/Akainu showdown. I ruled it out because of the 4 year gap when the fire/ice is not related to that incident after all.
Perona cover story, she could yet join the crew haha. No.
I think Big Mom may have an agency, like Crocodile had Baroque Works. We might get a saga-length build up to the inevitable battle between the Straw Hats and Mom's crew.
Perhaps the children are there to test some kind of vaccine for an illness rising out of the Vegapunk Incident. "We're not sick any more..." it's working? Their usage as experimental guinea pigs makes sense and would explain why they want to escape from this place.