Hi,
no, you can not get Unemployment benefits in the 6 months job-search period, which is included in your study-visa. Regardless of the fact that you can work certain hours and full-time in summer-months, its still against the terms/conditions for a study-visa to receive any kind of public assistance / benefit payments, including state educational grant (SU). I think this is also stated in your visa-document.
However, it is possible to get SU if you as a student meet a number of special conditions.
We do not give advice about residence/work-permit and therefore I can not say if the fact that you have a partner with permanent residence change anything. However I can say that for being eligible for Unemployment benefits (“dagpenge”) you must be able to take any kind of full-time job with one day notice. You can ask SIRI about residence/work-permit.
Best regards,
Anders Weber, A-kasser.dk