Would I receive dagpenge as a non-EU foreigner on a J-type permit?
first of all I would strongly underline that we can not - as we also write in the Q&A question page - guide about work permit. The only thing we can say is that if your work permit allows you to take job with one day notice, then it should also be possible to get dagpenge (assumed that you meet all other requirements for being eligible for dagpenge).
You must be covered by the Job Change Rule
The job change rule makes it possible for you to start your new job before you have received a new residence and work permit. All you have to do is submit your new application on the same day as you are going to start your new job at the latest.
Can you receive dagpenge, while on a job-searching permit?
In relation to being eligible for dagpenge or not, it is important (among other requirements) that you can start on a new job with one day notice.
So the question is if you are allowed to start a job with one day notice.
If you begin a job search stay (the 6 months job-searching permit ) and is covered by the job change rule, then you can start work as soon as you apply for a new work permit. This is done through SIRI.
You may therefore begin to work once you have submitted the application for a residence and work permit.
If SIRI has given you a job seeking permit because you have lost your job under the Fast-track scheme, the Pay Limit scheme or the Positive list, through no fault of your own, then you are also encompassed by the job change rule, and can start work as soon as you apply for a new work permit.
It is my opinion that a person who is covered by one of the two scenarios for the six months job-search period, is able to start a new job with one day notice, and therefor will meet this requirement for being eligible for dagpenge.
I am not able to say if you will be covered by the job change rule. This is something you must ask others about, for example SIRI.