I don't know about rentals specifically in Iceland, but you could check the usual sources: Airbnb, VRBO, etc.. I would suspect that, like most of Europe and Canada in July, there will probably be a one week minimum condition for most available rentals.
DH and I spent a whole week based in Reykjavik and loved it!! We had no problems finding things to do. We did several day trips, mostly as excursions. However, with a family of 4, you could rent a car and easily do it on your own (with a couple of exceptions, like the long drive south to Vik and a guided tour that focuses on the geology of Iceland).
Here are some suggestions:
One full day just exploring Reykjavik itself, including going up The Pearl.
One full day driving the Golden Circle on your own: Thingvellir, Gullfoss and Geysir, and/or include some of the Viking/Saga Trail as well.
Thingvellir Park can be its own full day trip, as it's great for safe hiking and even has snorkeling in July (Silfra fissure - we went in May, so couldn't do that).
One day with a 4X4 excursion exploring the unique geology of Iceland: lava flows, mud pools, fumaroles, glacier walk, caves, etc, - whatever interests you most. But it absolutely must be done with an experienced guide!
One day of combining a ride on the 'small horse, big personality' Icelandic horses (with farm lunch), with a lazy soak in the Blue Lagoon -- this day would be a big hit with the girls!
One of the days, you can schedule in a whale watching & puffin spotting boat trip.
One day heading south towards Hekla and Vik: Seljalandsfoss and Skogafoss waterfalls, the black sand beach and basalt pillars of Reynisfjara (very much like Ireland's Giant Causeway), the small folk museum of Skogar, and views of the volcanoes.
Here's advice that you cannot ignore: only go hiking where there are marked trails - the geology of the terrain is quite seriously dangerous (not an exaggeration!).
Remember: travel days don't count for scheduling activities.
Finally, here's my opinion: Paris is easy to get to for the girls to go back one day on their own or for you to return as a family (or as a mother-daughters trip). A quick visit for the 4 of you to Paris would be a good introduction for now.
A family vacation in Iceland would be priceless and a truly unique experience! I would tip the time and focus of this trip towards Iceland - do it once and well.
