I think you'll have a hard time registering your residency, given that a reason to stay is part of the documentation that link says is required. If you haven't got a means of support, they may not let you register as a resident.
if you're not registered, you don't get a national ID, which means you pretty much can't do much. No bank account, limited healthcare access, and pretty much everything else. Iceland runs on the kennitala.
of course it would be nice if someone from EEA could confirm this since I don't have personal experience. I just know that before my registration went through, Iceland was a very frustrating place to get things done.