1) As a start of point you need this "frigørelsesattest" for being able to get supplementary benefits. However you don't need frigørelsesattest, if:
- your employment contract states that you do not have a notice of termination. If you are a loose employee, you usually have no notice of termination, or
- your reduced working hours can be placed at any time so that the working hours do not prevent you from taking over other work. This applies to e.g. if you organize your work yourself and do not have a duty to perform it at certain times.
You have to be approved for supplementary benefits ("supplerende dagpenge") by the A-kasse, and its only possible if you meet the above mentioned requirements. Otherwise its not possible to receive dagpenge at all, while at the same time having work.
All of our answers to your questions is based on the requirement saying that you must be 100% available for the labor market. And if you can not start any full time job with one day notice, then you do not meet this requirement.
Anders Weber, www.a-kasser.dk