The Greek islands have beautiful weather in the months just before and after peak season. It’s a great time to see the islands, save money, avoid the crowds, and still have great weather (though not as hot as July and August). If you want to see the super-popular islands of Santorini, Rhodes, Corfu, and Crete without the tourists then this is a great time to visit.
Since airline reservation requests are often made by city-pair (such as "show me flights from Chicago to Düsseldorf"), an airline that can codeshare with another airline for a variety of routes might be able to be listed as indeed offering a Chicago–Düsseldorf flight. The passenger is advised however, that airline no. 1 operates the flight from say Chicago to Amsterdam, and airline no. 2 operates the continuing flight (on a different airplane, sometimes from another terminal) to Düsseldorf. Thus the primary rationale for code sharing is to expand one's service offerings in city-pair terms to increase sales.
Hydra is great but it does take a bit of time and effort to get to from the Cyclades. You’ll need to ferry to Athens, then might have to overnight there, then ferry to Hydra. Whereas Naxos, Paros, Milos, Folegandros, etc. would all be one direct ferry from Santorini or Mykonos. If you do go to Hydra then Leto Hotel is a great choice close to the port and shops and restaurants.
To sum up: a lot depends on your own stamina because island hopping means packing and unpacking, getting on and off buses and ferries. Limit your choice of islands to perhaps one or two less than you think you can manage. Maximise transport links to avoid backtracking or port-transferring and since you are traveling high season be aware that you will usually need bookings ahead at most places. It is possible to turn up on an island and not find a place to stay or have to make do with a third-rate option.
My boyfriend and I will be traveling to Greece on August 1-11th. We have 10 days. Is this a feasible itinerary for a couple in their early 30s who want beach, relaxation, good food, boating, and some history? Fly into Athens have one full day there then fly to Naxos for a day and a half, Milos for 3 nights, then Santorini for 3 nights, then back to Athens for our flight? We chose Milos over Naxos at first, but after reading your blog it seems the beaches in Naxos may be better?