I can only recommend a stay at beautiful Casa Spisone!
It's perfect for a two-couple vacation, because of the two seperate units with each their own bathroom and toilet. The place is also suitable for a family.
Both units have aircondition.
Do not get scared off by the "room with no windows", it's perfectly fine and even if it had windows you wouldn't use them - as you're trying to keep the cold inside.
We really enjoyed the terrace.
It was easy to reach Taormina center with the cable cars (3 euro per person), but I will recommend ordering a taxi from the house to Taormina center which is only 15 euro in total (in rush hour 20 euro). We received the number for the taxi station from our first driver (+39094223000). Also remember to mention one of the restaurants close by, as it is hard for the driver to know where the address is located.
The beach that the direct access leads to is ok (walk a bit to the left to avoid all the stones). I will recommend to go by car north where there's better beaches, we found a great spot at Lido Lime where the water was perfect and sunbeds were good priced (2 sunbeds with umbrella 15 euro all day, 10 euro half day), the food there is nothing special though. I also recommend paying visits to Alcantara riverpark (1.5 euro per person).