Several organizations (including the following) help asylum seekers with advice and assistance:
Dansk Flygtningehjælp (Danish Refugee Council: general counseling and help with Dublin-complaint cases)
Refugees Welcome (general legal counseling)
Trampolinhuset (lawyers and help to refugees who move to a municipality)
LGBT Asylum (help to sexual minorities in the asylum system)
Red Cross (assist rejected asylum seekers with return to their country of origin)
CePi (family reunification cases, not asylum)
You can get answers (in your own case or generally) from the Danish authorities by calling directly to the Danish Immigration Service, alternatively through a person with a written power of attorney, see Immigration Service's business hours.