FunBrain.com and poptropica
there is no website,you need a membership to play online and off.
Pokemon.com's funzone
No you have to get an iPhone or itouch and download the app
There are various websites where you can play chess. Below are two of them. www.chess.com , www.freechess.com
games2rule.com, Y8.com, and maybe miniclips.
Webkinz and Build a Bearville
dubhappy.com
duelingnetwork.com you can play other people around the world
As of now, no.
You will know if the website is legal depending on the content displayed on the website. If the website lets you download games for free, that you will have to pay a lot of money for at a store, then the website is not legal. But, if the website lets you play short online games for free, then it is most likely legal. A very good legal game website is GreatWebGames.com
No website that I know of has a full, legal version of FATE in a flash app.