For visitors to Denmark theres an interactive map where you can see lots of stuff such as free shelters and much more... unfortunately theres not an english or german version yet, so heres a fast translation of the main points in the green drop down menu. In the submenu theres icons that should help a bit.
Nature routes (car, bike, hiking etc.)
Facilities (water, toilets etc.)
Places to spend the night (Campsites for booking (usually free), primitive campsites such as shelters, free areas (state forests and parks I guess) to put up a tent.)
Things to see (nature, history, national parks)
Bathing and fishing (think u need a license to fish tho)
Sailing (harbours, kayak routes, canoe stuff)
Calendar of events.... you can input desired daterange.
As for the primitive campsites there might be a nominal fee... but some (think 720) are free shelters.