1. I would rent a house from Phuket for the duration of your trip and then take smaller trips to for example phi phi islands and maybe spend a few days to a week in bangkok too on either end of your trip. May to november is the low/rain season in thailand so rain might ruin your trip. i spent last may here and it rained every day but only like an hour a day. when i came back in july it didnt rain for a month and then it rained like all september-november but it could be rainy all the time during may-november or it could be sunny. lets just sum it up that it will have a lot of variance
While it is low season in may-nov there will be some/lot tourists in phuket. not sure how many is too many for you. now there are wayyyy too many here but it is high season so it is expected.
2. like I said I would get a house. the problem is that i assume you want everything to be hassle free and when you arrive you would have the house ready. but i wouldnt rent any house just because you saw some nice pictures on the web, especially if you dont know phuket. you can get internet to most houses and if your budget is above average then it wont be a problem with the house you are getting. if you would like me or someone here to help you more with houses/locations and stuff you can pm me your msn if you have one so i can ask more specific questions about location etc.
3. if you are fine with cheap thai food and dont need to be sitting at nicer restaurant all the time then you wont have big restaurant bills. you also mentioned cooking yourself which saves you some too. if you need pizzas or steaks then it is going to be a big part of your budget. you can rent scooters for 2.5-4k baht per month (roughly 70-110 usd or 50-80 euros a month)
4. atleast chili and propagandapoker work. havent heard of any skins that dont. afaik betfair and some(all) ongame skins are blocked but all other work. you can use vpn to get around it too
5. people are surfing, yes but i have no idea if phuket is decent for that overall
hope it helps. like i said pm me for msn if you want something more specific