Messenger question

You might not be able to block all messages, but you should be able to at least block messages from people who are not your friends (go to your privacy settings). I would also suggest going to your notifications settings as I suspect that's more of an issue. What do you care if the messages just pile up in your inbox if you never hear about them?

Me, personally, I have something like 99% of all notifications turned off and I think the only notification I get is if someone has sent a message in one of the pages I manage.

Ditto to what Jespah said. I have my personal page totally locked down and all notifications turned off. I only get a few specific push notifications from my business page.

It IS possible, but you'll probably have to do it on the desktop version of Facebook.