Moving to Iceland but not work there

I'm assuming you hold South African passports, which means it will be tricky to register as residents in Iceland. If you don't have a job, an Icelandic family, or plans to study in the country,there isn't a visa category that fits.

Without a visa, it is hard to be registered in the country, and if you are not registered, you won't have a kennitala. This ID number is essential for renting housing, borrowing books, and everything between.

I recommend you read through the Icelandic immigration site at utl.is to familiarize yourself with the visa options as an important next step.

I suppose the first thing to do is to make an appointment with the embassy in South Africa and see if they can help us

As far as I can determine, there is no embassy or consulate anywhere in Africa. (The list of locations is here: http://www.iceland.is/iceland-abroad/ )

Have you checked the immigration site I posted above? If you are dead set on getting info from a human being, you can try calling them during their posted hours. My experience with embassies is that they are not the best source of information on the more unusual cases anyway. The immigration authorities are the ones you have to go through if you want to reside legally in Iceland, so why not check with them first?

Regarding importing pets, it is an expensive and lengthy process due to the required quarantine times. More info in this thread: https://www.expat.com/forum/viewtopic.php?id=103062