yes it is important that you become a member of an A-kasse at latest 14 days after graduation, and in this way having your graduate status/rights registered in the A-kasse.
If you do that then you will have a right to start to receive Unemployment benefits during the following two-year period, without needing to meet further requirements. The right is only based on your finished education (graduate rights).
So you can easily go on vacation in another country (EU or non-EU country), and then come back to Denmark to start receiving benefits as unemployed.
You need to stay as member in the A-kasse and pay membership fee in the months you are traveling. But you don't need to keep a residence in Denmark to become/stay as member in the A-kasse. But the day you apply for Unemployment benefits you off course must have a residence in Denmark again.
Only if you go to another EU-country to work, then you can not stay as member of the danish A-kasse any longer.
About your question of when/if CPR status will be affected, you must contact other authorities. I think the best would be to ask your municipality of residence. You could also read more here:
Anders Weber, www.a-kasser.dk