Our stay at Fanis and Tina's cave house was perfect. The description is very accurate and the space is very clean. The views were absolutely picturesque and we couldn't have asked for anything better. We stayed in the cave house during the evenings mostly and during the day we drove to Perissa beach (a car is absolutely recommended) and explored Oia and other villages on the island.
We ate at a few restaurants in Oia and the one we recommend is Cadouni - it's a splurge but definitely worth it - best meal of our entire trip. Make a reservation at Santo Wines for a sunset dinner a few weeks in advance and you will not be disappointed by reasonably priced wine tasting options and snacks (they also have meals available) - this is a must (and the views are incredible). They also had live music the night we were there (they don't have this every night) which added to the romantic atmosphere. Make sure you take the time to see the sunset by the lighthouse (I recommend climbing to the flat space at the bottom closer to the ocean) and then you can replenish your energy by eating at the Good Heart (get the pork chops).
As for the cave house, the service along with the porter, the concierge-like help and housekeeping were the cherry on top. Communication was also super easy and we received a response right away. Everyone was so friendly and made our experience super pleasant. We can't wait to come back!