Now that everybody is aboard and interacting, there is a period where she ship will stay motionless until Fern finishes handing out pokemon. This means an influx of literal fresh trainers are going to be on the ship, as opposed to the handful who have at least had their pokemon for a week or more. 

As for battling, there are older trainers, 20-40 years of age aboard the ship who act as the mediators for the pokemon battles, as well as these folks there are one or two gym trainers, what gym trainers these are I will leave to you folks. These adults will intervene in battles they deem to be requiring an official setup.