With children, I personally think you will driving too much and too far if you live in Coral Bay, Peyia or Sea Caves. All activities are based in Paphos, Yeroskipou, Kato Paphos, Chloraka and on the road up to Polis mainly.
Parks for kids are by Debenhams in town, Yeroskipou beach, Aphrodite hills and by Chloraka (on the beach) as well as some small ones harbor front by the hotels. Kids the ages of your girls do dance classes a lot, music classes, go to movies, bowling etc all of which are in or around town.
As for being cooler, even further in the hills like Kathikas my wife still finds it unbearable. When it is hot, it really is hot!! I think you need to rent first and have a good look around before you buy. That said, I can suggest some good locations with kids - Tsada as PeteandSylv have already mentioned as well as Secret Valley and Aphrodite Hills, Konia, Armou, Anarita (mentioned by Anski), Tala, and Kamares for starters.
The ISOP is a good school and they are accustomed to taking in new children that have English as second language and/or new language so they should catch on fast.