There are many sites that allow you to embed their content. Otherwise, you will need permission to use them.
You could download the flash through their site and upload it to yours though I'm not sure if thats legal.
Well, you either have to hunt around their site for information on embedding the games, or they might provide a link on the game's page. Regardless, you can always contact them and find out if they allow it.
They generally make it pretty easy to tell, I know for sure http://freeonlinegames.com does, as they have the embed option right below each game. Also, I do have a small database of games located at http://wiredtron.com/games which you may use any of.
Ever heard of the word freeware? Just go hunt for freeware games and you can use the same games on your site. You can't charge money for it though. Some may list restrictions but most will allow you to display it on your site or have it for download.