Stayed at the Zante Star for 2 weeks (came back yesterday) it was fantastic, hotel is great and all the staff (Yannis, Tracey, Irena, Lia etc) were friendly, helpful and all a great laugh. Louis the dog is the best really great to muck about with. The hotel is extremely clean and tidy, our room (No.222) had the best view and also overlooked the pool. Location is OK, not to far from the beach and also within walking distance from the centre of Tsilivi (approx 10 mins). There is also a Supermarket (Alex's) just at the end of the road about 2 min walk. Would definately recommend the Zante Star and we are definately thinking about going back. No it doesnt have Air Conditioning but you can hire a fan from Reception and that does the job nicely.
In answer to your question, yes there are fridges in the rooms however I am unable to tell you where the nearest doctors or hospital is. When you arrive Tracey will be able to tell you, she is English and extremely helpful.