All properties that you find using the search function on www.boligportal.dk/find are available.
It is up to the landlord to remove the ad, once he or she has found a tenant. Even though an ad appears in your search result and the landlord has yet to sign the lease with a new tenant, he/she may be close to finding a tenant. To have an idea of how likely it is that the property is still available for rent, you can check to see when the landlord posted the ad. If it is a long time ago, then the landlord is more likely to be close to finding someone, than if the ad is brand new.
Thus, you cannot expect to be first in line for every property you find on BoligPortal.