Is there a PHP plugin or an open project offering a simple login/connection system to most of the major sign in providers simply by providing an API key then storing the linked account information in MySQL? Or is there a system that can be extended to include various other login providers as they become available?
It is not a plugin but something you add to your site. However, take a look at Dope OpenId, it is rather easy to use and handles most implementations for free and you are in total control and not dependent on yet another service.
Edit: I haven't checked if it handles all of the above, but I have most of them on my login screen and they work.