The Dome House is lovely and has all of the amenities you would need. We were a party of eight and we were never crowded; there is ample room in the home. The house is very clean and the inside and outdoor spaces are very well maintained. There are a couple of large screen TVs, at least one of which had access for games and cable/satellite TV. The kitchen is well equipped and includes all the necessities.
The house also includes a squash court and a nice sized pool. We would have, typically, 4 adults and 2 children in the pool with no problem and the pool is cleaned daily. The communication before and during our stay with Avtar and Nathalie was very good. They provide a very detailed guide to the house and surrounding area that is very useful. The caretaker, Bledie, is very nice and available if you need assistance.
You can walk to a local bakery (Family Bakery) and to the local supermarket. The walking paths to each are along busy streets so I wouldn't recommend a walk with children. There is a path along the caldera to the bakery but, again, we didn't feel comfortable taking that path with the children. With that said, car rentals would be useful especially if you plan to see other towns on the island, which you should. Getting to the other towns, with the cars, was quite easy. There is a winery (Venetsantos) that is within easy walking distance and another, Santos, that is near the supermarket.
Be aware that is can get very windy as you are at the top of the caldera. Things will be blown about but it's manageable once you get used to it.
Our family had a great time here and would recommend staying here during your visit to Santorini.