you can not receive Unemployment benefits and at the same time participate in full-time studies. And this is regardless of it is online or normal study where you meet at the school.
Some few studies are possible to follow when receiving unemployment benefits. But this is only part time studies, and when receiving Unemployment benefits you should always be available for taking full time job with 1 days notice. If you think your online study is under this category, then we recommend you to ask your A-kasse, if the they can accept this.
But the clear rule is, that it is not possible to study and at the same time receive unemployment benefits.
You also asked if you can seek job abroad?
Yes this will be possible up to 3 months while still receiving Unemployment benefits from your danish A-kasse. You can read more about this option here:
Please comment, if you need further information.