Do I have to travel every day with a continuous pass?
No, with a 15 days continuous pass (and any other continuous pass), you don't have to travel every day. However, you do have the freedom to go by train on each of your travel days if you want to.
The Eurail Global Pass is available with 15 or 21 days or 1, 2 or 3 months of continuous travel.
Read more about what 15 or 21 days or 1, 2 or 3 months continuous means