The discount applies in the countries on the list on the AVIS booking page and for short-term rental (1 to 28 days). Otherwise, the rules are the same as those that usually apply to car hire through AVIS.
Click to rent a car with discount through AVIS now (note in Danish)
With regard to insurance, you should note that your existing insurance policies with IDA Forsikring may provide cover that makes the purchase of extra cover in connection with car hire superfluous. We recommend that you look into this before you arrive at the rental location to pick up your car.