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My first post on the site.

I'll recommend using the site www.rejseplanen.dk for travelinformation using train and buses in and outside of Copenhagen. The site is Danish by default but can be changed. Please note that public transport isn't cheap, atleast not outside of Copenhagen. So travelling from Billund to Copenhagen can be quite expensive unless you are lucky finding cheap tickets.

I don't know much of the Nightlife in Copenhagen but i'm sure you can find several locations where there will be open to six in the morning.