Hi,
our answers of your questions (updated April 12, 2019):
You can be member of an A-kasse. BUT for you it is important if you can get unemployment benefits in case of unemployment. And in this matter it is important and make difference wheather you are from EU or non-EU. For being eligible for unemployment benefits as an non-EU you must (in your residence card) have permission to work FULL time in Denmark.
This because one of the requirements for getting unemployment benefits is that you must be able to take full time job with one day notice, and you can not. Your residence permit have some legal restraints (you only are allowed to work 20 hours/week).
Next, you must also be able to meet the residence requirement that came into force from January 1, 2019.
Therefore consider A-kasse membership for you would only be relevant if you meet the residence requirement and think you can get a work permit (which allows you to work full time) after your PhD. This could for example be an Establishment Card.
PhD students have two ways to get dagpenge:
For being eiligible for benefits as en employee you must (among others requirements) have been member for at least the last year, and have had at income of at least DKK 233,376 (2019) during the last 3 years (in total).
You can sign Up for A-kasse now, even though you are on maternity leave. However, remember that you also must have income (from work) for 12 months to fullfill the income -requirement. If you get paid maternity leave from your employer (like salary), then this can be used to meet the income requirement.
As graduate (this also include PhD) you don't need to meet the 1 year membership period requirement and income requirement. See more here: http://a-kasser.dk/non-eea-graduates/
Work permit/establishment card:
If you apply for - and get establishment card, it will include permission to work full time in Denmark. In this case it will give more meaning for you to apply for normal paid membership of an A-kasse.
Best regards,
Anders, www.a-kasser.dk