it is correct that you NOT can get benefits before 4 week after finish education (you lose unemployment benefits for a total of 3 weeks), because you have not been member for 1 year.
It is also correct that you as an EU-citizen can transfer your unemployment benefits to another EU-country (Spain).
The requirements for transfer is that you are "full" unemployed (not partly or periodically) and that you are entitled to get unemployment benefits in Denmark.
You can NOT be seen as unemployed when you have not finished study (study is compared to work in the danish social security rules/legislation) . So you can not register as unemployed jobseeker at Jobnet.dk before you are graduated (beginning of July).
About your last question:
Because you are citizen of Spain and you will go to Spain after graduation to search for job, you will normally could get the exception to the prior 4-weeks period of registering at Jobnet.dk. Please ask Akademikerns A-kasse about this.
When you use the graduate rules you get the 3 weeks quarantine.
This quarantine must be proceeded before you can start getting benefits. Are you newly graduate and unemployed, you may well go on holiday. Remember to report vacation to Jobcenteret no later than 14 days before the holiday starts.
Normally to proceed quarantine you must be enrolled in Jobcenter (jobnet.dk) and otherwise be available for work from day to day during the quarantine. However are you newly graduate you can easily travel out of the country in your waiting/quarantine period.
We have been inspired by these articles (in danish):
(Lærernes A-kasse, regarding graduates and quarantine period).
(CA A-kasse, regarding transfering benefits to other EU-country)