No, no pets are allowed in any of our apartments. If tenant breaks the rules it can result in end of contract and cost of painting the apartment.
The elctricity and heating cost are charged from HS Veitur or Veitur, depending on where you live. 
No you need to contact the phone compnay you wish to buy service from. However in the apartmens at Nauthólsvegur it´s included.
You apply on https://hms.is/en/loans-and-benefits/housing-benefit and on your contract is all the information you need. 
No after your application has been approved you can´t change it. Please contact the office for questions about the changes you need to make.

No you don´t need to register your papers to be able to apply for housing benefits.