you do not write what nationality you have. I assume in the following that you are from another EU country.
If you are from another EU-country we/you don't need to think about if you have proper working permit.
Because, when talking about A-kasse (membership, and getting unemployment benefits) the most important thing is that you have the right to work full time in Denmark.
When consider to be member of an A-kasse it is important to know that there are two main requirements before you can get unemployment benefits:
you must have had at least 52 weeks (1.924 hours) of work within the last three years and been a member of an unemployment insurance fund for at least one year. Only work performed on ordinary working conditions are counted in the 1.924 hours. I can not tell if your kind of paid work is on ordinary conditions - but don't think so.
I assume you have an ordinary contract now - and if thats true you can apply for membership.
You can NOT catch up for months by paying now.
If you are an EU-citizen then you can transfer rights (membership-period, workinghours etc) if you move to another EU-country. You can read more about this topic here: http://a-kasser.dk/transferring-unemployment-benefits.html