Well lucky you... I'm just back from one week in Greece.
Concerning the deal: I paid an average of 35 Euros per night, staying in a double room, which would be around 1000 for the month. Now you're getting a house, so personally I would consider that a good deal. At least good enough.
I'm not quite sure what you need a "home-base" for though. A month is quite a long time. Are you looking for contact to the local people?
In general the impression I got of Greeks was that they *do* try to sell you things you don't need, as long as you don't say what you want clearly. *But*, if you know what you want, you will get exactly that. I would believe that old lady.
About the day trips: To get to the closest islands, you will be on the boat for at least 3 hours, one way. And some of the most beautiful islands (e.g. Santorini) are over 7 hours away by ferry. So I would spend at least two or three days per island. Have you considered a round-trip of the Cyclades for instance? What we did was rent some small motorcycles (10-15 Euros per day) and just drive around. That way, you get to see a lot of things -- most islands are rather small -- and enjoy some of that Easy Rider feeling.
My advice, but this is up to you of course, is to not make too big an agenda. Two or three countries in a month is quite enough for my taste, if you want to avoid just running from one tourist attraction to the next.
Istanbul is an amazing city. Go there
A last remark: You don't buy boat tickets at the port, you go to a small local travel agency and buy it there. (Most of the time, they'll try to sell you transfers, from the port to the hotel, and hotel rooms. But they won't try to rip you off, at least in my experience)
Any more detailed questions?
Hope this is of help to you...