Hi,
1) yes, if you are entitled to unemployment benefits (see below), then you can only start to receive benefits after 1 month quarantine, because you have not been member of a A-kasse as a student for at least 1 year.
2) I would definitely recommend MA A-kasse
3) The fact that you will move to a new address inside Denmark is no problem. As long as you are registered in the "Folkeregistret" with your new address, then the A-kasse will not ask for further proof/documentation. The A-kasse will be able to check if you are registered with a Danish address.
4) There are different ways that the A-kasse can get information/proof about your residence in EU (Greece). When you apply for unemployment benefits they will try to contact the public authority in Greece (assumed you have accepted that the A-kasse can do that). If they get the information from the Greece authority then everything should be fine.
If there for some reasons should be problems with getting the information from the Greece authorities, then you have the right to prove your residence in Greece by other ways. It could be with work-contracts, tenant-agreements etc.
Best regards,
Anders Weber, www.a-kasser.dk