Does plugin exist to allow Facebook login to comment and…?

Hello!

I’m not a WordPress expert but have been figuring most things out on my own. But, I’m having trouble trying to find a plugin that does what I want. I have searched and I have spent close to 5 hours looking so far and installing plugins and no real luck… I think it is because there is just so much out there that I’m having trouble narrowing it down and am hoping you experts can point me in the right direction. ??

There are two things I’d like to do and this can even be two different plugins if necessary.

1) I’d like the visitor (non registered user) to the WordPress site to be able to login, using their Facebook username/password/login and make a comment on the WordPress site without the comment needing moderation BUT, still allow other visitors to make comments via the normal WordPress way that still require moderation. Basically, if they are willing to use their Facebook account (not annonymous) then they can make a comment and have it show up.

2) I’d like to be able to post an article to my WordPress page and have this automatically post to a FaceBook PAGE and, if possible, also have it so when a Facebook PAGE has an update by the administrator of the page, have it automatically post as an article on the WordPress site. Pretty much have it so no matter where I post, it goes to the “other” site. If this isn’t possible then I think I did see a plugin that allowed me to click a checkbox to “Post to Facebook” but all be darned if I can find it now ??

Thanks for any recommendations you can provide. I’ve installed about 10 Facebook plugins already and no joy. I thought Jetpack might work but for some reason when I installed it the “Sharing” part of it tells me I don’t have authorization to access the page (I read a recent update was supposed to fix this but it didn’t for me) so I’m just trying to do it with a separate plugin. If Jetpack is my best option I’ll try to figure out why the sharing section of it doesn’t work.

Thanks!