You can camp in Denmark, there are a number of free tenting areas around the country. See
http://naturstyrelsen.dk/naturoplevelse ... teltning/#{"x":566067.2000000001,"y":6314393.6,"zoom":2,"categories":["9"]} unfortunately most of the information is in Danish. There are also many "shelters" available through out Denmark
As for tours, take a look at West Coast trail
http://naturstyrelsen.dk/publikationer/ ... -bulbjerg/ Jutland (Jylland) is your best option for longer walks, especially along the coast. However, in July and August there are many car based holiday makers from Denmark and Germany so it is busy on the West Coast.
This link (in Danish) gives a list of the walks (vandreturs) in Denmark.
http://naturstyrelsen.dk/publikationer/ ... rsfoldere/As well this link provides a list of routes (vandring) (some with GPX files)
http://www.friluftsguiden.dk/?activity=vandringTravelling around Denmark as an english speaker is not a problem and the public transport system is efficient and reliable, albeit expensive in AUD terms. You can plan your travel on public transport using
http://www.rejseplanen.dkBTW I am an Australian living in Copenhagen and have hiked a lot in Denmark see
https://nielsenbrown.com/denmark-wanderings/ for some of walks I have completed.
I am happy to answer specific questions, once you determine what and where you plan to go. Enjoy