Hello Mart, this is Beardy again, normal service has been restored!
No worries about the igloo thing, they don't mind having a bit of a laugh at themselves from time to time! I think the decision to not constrain yourself to the roads is the right one - there is a LOT of distance to get to some places here (which may be why many Icelanders go for various wagons and tent trailer affairs) but there is a lot more to this country than just the roadside.
But, something you will have to think about is how to get from A to B when B can be rather far away and public transport outside of the Reykjavik capital area can be... patchy, to say the least! I've managed to hitchhike from Rvk to Akureyri in one day (not bad I thought) and I think on Route One you will be okay but on the smaller, less often travelled roads you may have problems - we intended to go to the Westfjords but ended up in Akureyri since no one was taking that road! Other travel points: the fly bus (2000 Icelandic Krona from outside Keflavik airport arrivals last time I took it) will get you to Reykjavik, and Flugfelag Islands (
http://www.flugfelag.is/) and Flugfelag Ernir a.k.a. Eagle Air (eagleair.is) can get you from Reykjavik city airport to a number of sites around the country at a reasonable price (pay in ISK with Ernir, it works out a lot cheaper than paying in Euros or Pounds) compared to the patchy bus connections, and a lot faster. If you plan it right it can actually be pretty swift and doesn't require you to overnight in Reykjavik, should you not want to. You may also need to think about ferries depending on where in Iceland you are going to (certainly for the Westfjords and the Westmann Islands, or even if you fancy sailing here from Denmark or Norway with a stop in the Faroes). Also, there are a plethora of guiding/travelling/excursion companies here that might be going somewhere near where you are heading and have a spare seat.
I will ask ref the campsite list, but most villages seem to have one, usually near the sports hall or swimming pool so the showers there are used for both purposes. Also remember that in Iceland you have the Everymans right (I seem to remember it being called such?) to camp most places, and there can be natural 'hot pots' (pools from geothermal springs) that can serve as a good place to freshen up for free, and usually in great surroundings.
Speaking personally, I haven't done much overnighting out here, although I have done a fair bit of work outdoors and on the water. For September I reckon conditions will be quite changeable for you. I have been working outside in the South in August and been in T shirt and getting rather hot, conversely I have been hiking here in October in the North and almost gotten frostnipped hands! September in Iceland certainly isn't frozen, you will be far more likely to be on the receiving end of rain rather than snow in most areas. There again, the weather is pretty unpredictable here - there was 16cm of snow in May in Rvk - and some places here seem to have their own little microclimate - Akureyri in Eyafjordur for example is regularly colder and snowier than you might expect, even for its Northern setting. Wind in Iceland is a different matter to UK conditions too, the terrain seems more open, rolling, more 'big country' and theres a lot less trees to break it up. That and the fact that a lot of places in Iceland are near the coast makes this quite a windswept little island.
ICE SAR and the Coastguard have certainly come across as very professional when I have met them (not on the receiving end yet, touch wood!) and I would recommend touching base with them if you are going off the beaten path, unfortunately every so often bad things happen, even to locals, and mobile phone coverage here can be unreliable, so better safe than sorry. Do you have any ideas about where in Iceland you want to travel to?
I realise this is turning into a bit of an essay now and I've drifted a long way from tents vs bivvys so I will step away from the keyboard and wish you all the best with your trip, this is a great country and I'm sure you'll get a lot out of it.