I have Facebook comments social plugin integrated into my website. I was wondering if there is any way that user comments posted on my website would show up on out fan page as well? Something like User XYZ has commented on article ABC?
Sounds like custom functionality to me.
https://developers.facebook.com/docs/reference/plugins/comments/
How do I know when someone comments on my site?
You can subscribe to the 'comment.create' and 'comment.remove' events through FB.Event.subscribe.
And that process is here:
https://developers.facebook.com/docs/reference/javascript/FB.Event.subscribe/
Create an app publish an action, on behalf of the page (NOT the user). But here's the kicker: if they don't have control over the privacy of it, you probably shouldn't use their name on your page wall, at least not their full name.