I did a ton of research planning a trip to Greece I went on a year ago, so here are the highlights in regard to your questions:
1. Look into the itineraries fairly closely before booking anything. Many hit great ports, but you are only there from 8-noon, or 1-5. With loading and unloading the ship, that's hardly any time at all to enjoy a place the way you should. For this reason, we chose not to do a cruise and to spend 3 days or so on three different islands in Greece.
2. It's my understanding that American cruise lines offer greater service, and usually better food, than European lines. You'll pay a little more, but for me it would be worth it. If you're looking at the European lines, Costa has some very good itineraries and they have sales occasionally (I'm somehow on their email list)
3. Time of year - May is great - very nice weather, but not stifling hot and the crowds haven't descended yet. The Med gets super busy in the summer (June, and especially July and August), to the point that it's not even enjoyable with all the throngs of people. September would also be a nice time for the same reason.
I agree with checking out cruisecritic.com to get more information.