1. It is a very good idea to become member of A-kasse with graduate-status. And this have to be done at latest 2 weeks after graduation. In this way you have the right to start to receive dagpenge up to two years after graduation. And you do not need to meet the normal requirements as a wage-earner. You have the right as graduate.
Furthermore with 3 months work (at least 320 hours work) after graduation, you can get your dagpenge-sats (the amount you receive in dagpenge) recalculated based on your income in the 3 months. Normally this will mean that you will get the highest dagpenge-sats, from January 2021, if you have to go on dagpenge from there (and not only the graduate-rate).
2. Yes you can get dagpenge from January 2021, if you follow the recommendation in my answer for question 1. And remember to ask the A-kasse to recalculate your dagpenge-sats based on your income from the job.
Let me know if you have further questions.
Anders Weber, www.a-kasser.dk