I know both operators have services from Copenhagen Central Station to Copenhagen Airport. Question: can I swipe in on the Rejsekort app and then take whichever train comes first (DSB or Oresundstag)? Or do I have to decide in advance which company to go with and get a ticket unique for a certain operator?
Public transit is fully integrated in Copenhagen (this is quite common in Europe). So the ticket is good for a trip from the airport to thje centre, on any mode of transit. So wether you take a DSB, Oresundtag, or even the metro it does not matter.
A single ticket is valid for a number of zones, and a certain time. In this case you need a 3 zone ticket, and it is good for 1 1/2 hours. Not that this ticket allows you unlimited transfers, so you can continue on the metro or bus or S-Tog to your final destination, as long as it is in Central Copenhagen.