How to setup Easyxxx/Facebook interactions?

Hello to the Dream Team

I would like to create more interactions between our site and our official FB page.
And I am wondering what kind of 'link', on both sides, I can set up.

1. From Easyxx => FB
- do we have any mean to publish directly to FB?
- any others tips?


.2 from FB => Easyxx
- I suppose that I can publish from FB a kind of permalink to a group, a blog's publication, an event?
- any others tips?

Thanks for your help