You can only get dagpenge, if you are allowed to take any full-time job with one day notice, without you need to apply for a new residence-/workpermit.
We do not guide about residence/work permits.
However, your current residence card seems to be a study residence card. This does not give you right to work full-time, and furthermore it probably only allows you to stay in Denmark until you finish PhD studies.
It may include a 6 months job-search stay, but in this job-search period you are not allowed to work, only to search for job.
If you have been granted a residence permit under the Establishment card scheme, you do not need to obtain a work permit. A residence permit under the Establishment card scheme gives you the right to take up employment without it being necessary for you to apply for a new work permit if you change your job/find a job.
However part of the conditions for Establishment Card is that you do not receive unemployment benefits for new graduates (dimittenddagpenge) or any public assistance under the terms of the Active Social Policy Act (lov om aktiv socialpolitik) during your stay in Denmark. Unemployment benefits after the rules for salaried individuals is not forbidden for Establishment Card holders.
Since you have paid for A-kasse membership, and have paid job (salaried PhD Employee) you should be able to get dagpenge after the rules for salaried (not new graduates). But again - you can only get dagpenge if you have a residence card that allow you to take any work with one day notice. I think Establishment Card is enough - but you must get this verfired for example by SIRI, or others who guides about residence/work permits.
Like others you must meet the general requirements for being eligible for Unemployment benefits ("dagpenge").
Anders Weber, www.a-kasser.dk