The high season is a lot more expensive than the low season. I've just looked at a hire-car company that we've used before in Iceland (I've been nearly 10 times in the last 6 years with work), and the rental price goes up by about 50% a day in high season compared to low. The exact amount depends on which car you go for.
The high season usually extends until the beginning of the last week of august be varies between companies.
Finding activities you can usually improvise whilst there - and in fact I would suggest for some things it would be better as they are weather dependent. However, savings can usually be made booking in advance.
Accommodation varies, if you are flexible with where you stay you can probably get away with improvising if you go off season, high season the approach is a bit riskier as places are generally busier. It depends slightly on how many of your there are in your group.
Coming from the UK Iceland seems very big and spacious, and once in the south east it doesn't feel crowded at all. Obviously the big tourist attractions like geysir, gullfoss, the blue lagoon, snow mobiling etc will always have lots of people, but for just hiking you'll probably be ok.