WordPress plugin as an webapp

I developped a website with Symfony2, and this website has an authentification, and you can create questions/answers to play, with statistics, etc.
I would create a wordpress plugin/widget, to easily include in wordpress website, but, as I don’t know anything about wordpress, I would have advices and links to good tutorials which include this kind of processes.

So the aim is:

• To easily include into wordpress a plugin to play the questions.
• The plugin will support a personnal authentification (like the facebook “like” button on a website, if you are not connected, a login popup appears, and then I guess a cookie will remind that you are authentificated)
• And then on the wordpress website, there will be, or full pages, or widgets, which will do calls with my server to display questions adapted to my user. (I don’t know the difference and the possibilities yet, don’t hesitate to tell me about)
• I am creating my webapi which use JSON to communicate the datas.

If anybody have any advices, and good methods to answer to this problem, don’t hesitate to share it on this thread. Thank you by advance.