So, when you're in Denmark, you can not register on any of the local carriers, even though your phone "sees" all the signals? When you're in Sweden everything works as it should; calls, text & data. Denmark and Sweden are both "Simple Choice" countries. I can't think of any reason this would happen except some problem at the HLR. The only suggestion I would have made is to reboot the phone.
T-Mobile has a direct number for international roaming support (+1-505-998-3793), which is free from your T-Mobile handset. I'll call them and report the situation.
Another example of why international travelers need a communications "Plan B".