Start 1-2 am, but Adams Peak starts the season 24.12.
Avoid night trains in Sri Lanka.
There should be direct trains.
The connections are very good.
You need to take a bus from Billund to Vejle,and then atrain to Copenhagen.
Let the hotel book a taxi for you. Which hotel?
Denmark is a safe country. The refugees mostly continue to Sweden..
You will be responsible for the driver's accommodation and food.
The airport is approx. 7-8 km from the airport, but there are trains available.
The seas will likely be too rough for swimming.