as you mention your self it is ok if you have stayed with friends or Airbnb. You must expect that the A-kasse will ask you for some documentation/proof.
I assume it will be easy to document that you booked stay through Airbnb. If you can document that you booked at least 1 day with Airbnb, and that day was very close to study-start, then I would think that will be enough proof.
If they ask for more documentation, you need to proof the stay with friend in other ways. If your friend can confirm that you stayed there I would think it could be ok, but I am not sure.
Any kind of receipts you have from the period may be taken into account.
However we can not exactly say how much the A-kasse will ask for.
Anders Weber, www.a-kasser.dk