here in Denmark employers never pay for employees' membership of an A-kasse. If he/she do it will be seen as income for you and you have to pay tax.
Also you will only be able to deduct the dues in your tax-balance if you pay yourself.
If your job is seasonal you can still be member of an A-kasse. But it maybe will be difficult for you to meet the requirement of have had at least 52 weeks (1.924 hours) of work within the last three years to be eligible for unemployment benefits.
About last question: It is up to your employer if he want to give you job again after period with unemployment. There is no rules saying that it is not possible. But according to your rights to unemployment benefits you must not "reserve" your manpower for an specific employer. In this case your A-kasse will not accept you as being free to take job from day to day. This also is a requirement for getting benefits.
So if you will get benefits after a season-job is finished you must search/apply for other jobs with other employers. You can not just wait until next season starts and start with the same employer again.
In genereal about seasonal work and unemployment benefits: Do you have a contract, it is always a good idea that there is an end date in this, because then the A-kasse can see that you really are unemployed after the end date.
Do the employer want to fire you before the season or the agreed period expires, be sure to get a written termination with applicable notice.
I have a job contract and it is terminated. But how can i prove to Unemployment fund, that i am looking for a job for real? And if i lose my terminated job, for how long i am going to get my benefits?
I am not sure why you want your employer to pay dues for you. It will be expensive for you because you can not deduct the amount in your tax.
But off course if another person (your employer) want to pay for you he/she can do it - but again it is never used in Denmark and it is an expensive way to do it for you.
With membership of an UNION it is different - the dues for UNION membership is normally calculated from you salary and paid by the employer. But here we talk about A-kasse.
You must be aware that UNION and A-kasse is two different things.
If he calculate it from you salary or do it another way dosen't matter. But if he says he pay your dues you have to be sure/check that it is actually paid to the A-kasse.
About proof for looking for job: If you get unemployment benefits in Denmark there is a "big" electronic system (jobnet.dk) where you have to register all your activities concerning searching for job. You must search at least two jobs every week. if you don't do that the A-kasse will stop paying benefits.
For how long can you have benefits? You can get benefits for up to maximum two years.