After july 8th is a good time. I made a list of places to visit, and realized that to get to the most interesting of those would require at least 7 to 10 days.

Majority of those places are in the unpopulated interior where it is most practical to sleep in huts like those:

huts or organized trips.

These huts have sleeping bag acommodations and some shared cooking facilities. There are almost no restaurants in the interior.

In the populated areas there are many Icelandic farm holidays.

I do not know how much driving and/or walking is suitable, access to the interior is through primitive tracks that usually require 4 wheel drive and good ground clearance and some of the places require some hiking.

Some of those places are very remote, we may go a whole day without seeing anyone else, even in july.

Below are picture links suggested by Einar.
Yes
Yes
Yes
Yes
Einar's secret place
Yes
Einar's secret place
Yes
Yes
Yes
Yes
Yes
Einar's secret place
Yes
Yes
Yes
Yes
Yes