I wouldn't call it luxurious, 3-4 star at best, but we've just come back from the Blue Lagoon Resort in Kos and it would certainly fit the bill for your other requirements.
Lots of activities/games/sports - every 30 mins from 1000 to 1700.
Kids club 10-1200 and 15-1700
Small water park
Largest swimming pool in Europe
Lovely grounds/Spa/adult only areas.
Extremely clean, good food, lots of staff. Beach is a bit of a disappointment though.
Best of all it was only 10 mins in a taxi to Kos Old Town, which fulfils the cobbled streets, cafes, history part of the brief.
We spent a few nights sitting watching the luxurious yachts parked up next to the fishing boats as well - how the other half lives 
You can do day trips to Turkey on a simple passenger ferry or hydrofoil. Or you can go on organised trips to other Greek Islands/dolphin coves/historic monuments etc.
(yes there are migrants, no we didn't see any trouble or begging, no we didn't see 60k of them, nor did we see any tipping up in rubber dingys)
There is also the more luxurious Blue Lagoon Village at the other end of the island that was lovely (we spent a night there) but it was too isolated for me.