Hiring a car and driving around is a very normal thing to do.
We don't know what is possible on the Icelandic roads with a 'normal ' car. We haven't decided on a car yet, but we've read that a 4x4 isn't necessary in September.
A 4wd is potentially useful in September if you want to travel the mountain roads, as they should mostly remain open, though the remoter ones can close in early September, and in some recent years widespread snow has come as early as 9 Sept, though on the whole that's not very normal. But there is an extensive network of roads that are open to 2wds.
Almost every village has a campsite, and there are quite a lot at famous country spots too. In remoter parts wild-camping is reasonably straightforward too, so long as you aren't in a (clearly marked) reserve. Though a few may close down after the end of August if they are in a school field or something.
There are a lot of itinerary discussing threads on the forum. It rather depends what you want.