I don't know enough about flying between Greece and Turkey via Cyprus, but given the political situation/divided territory it does not surprise me that there are no flights that allow for this.
I certainly understand not wanting to miss Cappadocia. The west coast of Turkey and Istanbul would be easier to see in your case, but I think Cappadocia is a more unique place to visit. Although I think it would be more restful to see just Cappadocia and Istanbul, your schedule is possible in six days if you fly early/late and rent cars. You will use about 1.5 of your 6 days for travel.
Unless you found a direct flight from Athens direct to Denizli, you'll have to come through Istanbul to get to Pamukkale. From Athens I’m guessing that it will take about 4 to 5 hours (two short hops with a short layover in between) plus another 20 minutes of driving. You’ll probably also have to come back through Istanbul to get Kayseri or Nevsehir (the two closest airports for the sites in Cappadocia). Figure another 4 hours for two flight segments/layover plus a 25 to 50 minute drive. Nevsehir is closer to Goreme which has more hotel and restaurant options than some of the other towns in the area. The flight back to Istanbul would take about 1.5 hours of flight time plus travel time into the city. Add in a bit more to the travel time if you’re planning to check bags.
Under than plan your trip would look something like this:
Day 1 = Flights to Pamukkale, ½ day in Pamukkale.
Day 2 = ½ day in Pamukkale, Flights to Cappadocia
Day 3 = Cappadocia
Day 4 = ½ day in Cappadocia, Flight to Istanbul
Days 5 and 6 = Istanbul
Given the time contraints above, I'd spend my time like this:
1. In Pamukkale divide your time between the travertine, pool, Hieropolis and the museum.
2. In Cappadocia see Goreme Museum/Buckle Church and nearby fairy chimnies, Uchisar castle, Kamakli underground city, and Sarihan caravansary/Avanos village. If you have extra time add on the Church of St. John in Gulsehir, Eski Gumusler monastery in Nigde or the Devrent Valley.
3. In Istanbul see the Topkopi Palace, Aya Sophia, Hippodrome, and Blue Mosque. This will take one very full day. Use a guide book based on your interests to fill out the second day.