In 2001 we tried to get to Askja at the end of July (24:th?) and the road was just cleared the day before, 2 meter high snow banks on both sides of road, night temperature was near zero. So bring double mattresses for the tent and decent sleeping bags. It might not be a bad idea to split the trip into 2 parts, first "normal" ring road with cheap normal rental car, then selected tougher inland tracks with a good 4x4, as the rental fees for good 4x4s are high. We were 8 people (parents and 6 kids 8 to 15 years old) with our own Land Rover Defender shipped from Bergen, living in tents (mountaineering type recommended, as the wind velocities are high enough to rip family camping tents apart). I do not remember anybody being cold anywhere, but we are Scandinavians used to subzero camping... Several times onlookers commented on our campground performance, 10 minutes flat from unloading the roof rack to pitched tents without any shouting or commands from the adults... Iceland is a place I will go again (5 times so far), amazing photo opportunity, everywhere.