Hi,
1. yes, if you meet the requirements (1 year membership and the income requirement) you will eligible for Unemployment benefits (“dagpenge”). As a swedish citizen you dont need work permit, so you can also be available for the labour market.
2. You dont need to sign up at Jobcenter (or online at Jobnet.dk) before you apply for “dagpenge”. You must sign up from the day you apply for dagpenge from A-kasse.
I assume that you have no “A-kassa” historic from Sweden. If you have been member of A-kassa in Sweden (or compulsory basic insurance in ALFA-kassen) before you started to work in Denmark, you could have transfered your insurance period from the swedish A-kasse/ALFA-kassen to the danish A-kasse. In this way you could probably already now meet the requirement of 1 year membership.
If you have not done that, then its too late now. It has to be done within 8 weeks.
Best regards,
Anders Weber, A-kasser.dk