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Are there some more restrictions about the place I'm staying for the quarantine?
I couldn't find an explicit list of requirements for Iceland but most quarantine/isolation guidelines strongly encourage a private room with a private bathroom. The government does publish a list of hotels that welcome guests in quarantine here: https://www.ferdamalastofa.is/en/about-us/novel-coronavirus-covid-19-general-info/accommodations-that-welcome-guests-in-quarantine
can I get the results in some paper form so I can present them when coming back to my country?
For the entry tests, you only get the negative test results in the form of a text message. If you need a negative COVID test for your return travel, you can order & pay for one. https://www.covid.is/sub-categories/trips-abroad
how does the quarantine look like if the second result is positive?
If you test positive for COVID while in Iceland, you enter isolation. Instructions for isolation can be found here: https://www.covid.is/categories/how-does-isolation-work & https://www.landlaeknir.is/servlet/file/store93/item39066/KSJ%20Instructions%20for%20the%20general%20public%20on%20home-based%20isolation%2004042020.pdf In short you continue to stay at your place of quarantine for another 14 days.
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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