you have to add facebook to your accounts (menu -> settings -> accounts -> add account) if you want it to sync contacts.
the facebook app does have push notifications; you have to open the app, hit menu button , settings -> refresh interval (never). then, make sure that in data delivery (battery & data manager -> data delivery ->social applications), you have automatic (push) selected.
i'm not sure how this affects battery though. i'm going to keep playing around with these settings to get an optimal life