The program I am going abroad with requires that I have a cell phone. There is a way for me to get a Danish SIM card or a cheap cell phone there, but I would rather use my iPhone if possible. Should I buy a SIM card and put it in, or can I use it as it is now? Or would it be safer to just buy a Danish phone?
Your 5S will work in Denmark. You can purchase a Danish nano SIM card when you arrive and just stick it in the iPhone for local service. You will then have a Danish number. You can put your Verizon account on suspension without billing until you get back.
Yes, you can use your phone as it is now and simply add the international calling plan to your line.
It will be MUCH less expensive to simply purchase a Danish SIM card to use in your phone when you get there. The prices for calls/data will be MUCH less expensive if you choose to do this.