1. The Insurance fee is collected by the 24 A-kasser from their respective members. The montly memberfee include insurance fee and administration fee. It is also the A-kasse which pays unemployment benefits, however there is a refund scheme between the danish state and the individuel A-kasser, so that the A-kasser are compensated for the net deficit they have.
The individuel A-kasser are non-profit. So the A-kasser should only have their administration costs covered - not more.
2. "Arbejdsmarkedsbidrag" (The labor market contribution) was introduced in 1994 to impose on the employees a part of the expenses previously imposed on the state in connection with, for example, expenses for unemployment benefits, activation and training of unemployed persons.
Therefore the income from this "tax", does finance both insured persons (unemployment benefits) and persons who are unemployed but not member of an A-kasse. If you get unemployed, but is not member of an A-kasse, then you can get a cash allowance (in danish: "kontanthjælp").
Administrationsbidrag also finance "kontanthjælp", and expenses to activate and educate unemployed persons in general (both insured and not insured).
When it was introduced back in 1994 there was a high unemployment rate in Denmark - and it was only meant to be a temporary tax.
But we have been paying the 8% all years since then.
In periods with low unemployment, like now, the danish state will make profits of the Unemployment Insurance Scheme.
So you can say that "arbejdsmarkedsbidrag" is like any other income tax.
3. I have not heard that the employer contribution is going to disappear. If you have any sources saying that, please let me know, and I can investigate further if you like.