Visiting Iceland: Details on quarantine regulations

I'm planning to visit Iceland for 1-2 weeks. My flight arrives on Saturday. So far I'm convinced about this few things:

I'm tested already at the airport (unless I arrive past 10p.m., but that shouldn't be the case),
I'm obliged to go to a place of quarantine where I spend 5-6 days,
I'm notified about the test results the next day,
after the 5-6 days quarantine I'm taking another test which results in my being let free to roam the country

The few things I'm missing are:

are there some more restrictions about the place I'm staying for the quarantine?
can I get the results in some paper form so I can present them when coming back to my country?
how does the quarantine look like if the second result is positive?

Have you recently been through this situation? I know the regulations could've recently changed. I'd appreciate any more suggestions about this process.

Best Answer

For anyone of you wondering the same thing, I've done some research and basically these two resources (prepared by Department of Civil Protection and Emergency Management, and Iceland's Directorate of Health) which describe situation in detail, were very helpful. (Although I admit I'm still missing details about the proceedings after the both possible outcomes of the 2nd test)

For everyone else - stay home, stay safe