I think we can give you an answer which covers all 4 questions:
You work as an salaried PhD and when your contract ends, you will have been paying to A-kasse for more than 1 year.
We assume that your income in TOTAL during the last 3 years - at the time you apply for benefits - will be at least DKK 223.428.
If this is correct, you are entitled to unemployment benefits when your contract ends.
For receiving benefits you must stay in Denmark. The only exception to this is the possibility to go to another EU-country to look for work and continue to receive your unemployment benefits from Denmark.
This you can do for 3 months with the possibility of extension up to 6 months.
To be sure to have right to get benefits after 3 months jobseeking in another country you must be sure to return to the country paying your unemployment benefits (Denmark) before or on the day your entitlement expires.
To transfer rights when looking for job in another EU-country, you also need to fullfill some special/other requirements, which you can read about here.
If you dont want to receive benefits even if you are entitled to, you don't need to register at all. This means you dont need to tell Jobcenter/Jobnet.dk anything, and you dont need to tell you A-kasse anything.
As long as you choose not to receive any benefits you are free to do what you like, and be in what country you like (EU or not EU), and you dont need to have adress in Denmark. And this will have no effect on your rights to receive benefits when you return to Denmark.
There is no restrictions on how long you can stay abroad, but be aware about the requirement about having income during last three years for at least DKK 223.428 in total, at the time you apply for benefits.
Off course you must continue to pay to A-kasse during your stay ("holiday") in another country. And when you apply for benefits after returning to Denmark you still must have had income in TOTAL during last 3 years for at least DKK 223.428. And off course you also must have adress in Denmark and live in Denmark if you want to receive benefits.
So, you can - and must - wait to register as unemployed at Jobnet.dk until the day you really want to receive unemployment benefits and are living in Denmark.
There will be no restrictions on the period of time that you have to be living in Denmark before you can get unemployment benefits, as long as you meet the mentioned requirements.
If you get a job in another country you must ask your A-kasse, if you can continue memberhip with them. Normally you are covered in the country where you work, and you can only be covered by unemployment insurance in 1 country at a time. We have been writing an article about "A-kasse and working abroad", but unfortunately at the moment only in danish language. But you can find it here.
As long as you are posted to another country for up to two years, for instance as part of a PhD/job contract you can continue membership with a danish A-kasse. However normally this requires that you or your employer apply for a special document from Udbetaling Denmark giving you right to keep danish social rights while working abroad.. We have been writing about posting to another country in this article. We are not sure if this also applys for PhD, but we think you should ask your employer.
Please comment on this, if you need further help.