If you read around the boards, you will see MANY discussions on you questions.
It can get quite hot in June, July and August. Given your limitations, I would go as early as possible in June. Don't forget to allow a few days in at least your departure port, like Rome, to see the fantastic city and get over jet lag.
No mass market cruise line does it any better than any other. If you like the product on Celebrity or NCL, start there with your research. Keep in mind that most Med ports are incredibly tour intensive. In other words, you are likely going to be doing a lot of 8, or perhaps longer, tours...day after day.
Your target area seems to be what is commonly called the Eastern Med. There are more great ports there than you can see in one cruise, so you can hardly go wrong with whichever itinerary you chose. There are many history and cultural heavy ports, along with some stereotypical Greek islands (like Mykonos or Santorini) and other sites to see.
You buy when you can...there is no magic time to book a cruise. There are sales going on, so you probably need to buy when you find what you want, and a price you can work with.