The email Ars received doesn't mention DMCA, so it's likely not the case. If it were, there are specific requirements that Facebook would need to follow including allowing Ars to file a counterclaim which then reinstates the content and absolves Facebook of liability -- the third party would then need to follow-up directly with Ars over their infringement claims.

If it's a valid notice, they have to take some action to stop infringement. It seems they should verify beforehand, but I'm sure they don't want to dedicate resources to that, so the easy thing for them to do is to lock or block access to the content and put the burden on the end-user.

Yes. It's similar to when YouTube pulls a video temporarily over copyright claims. I don't see how a page can go down several times, though - there are only so many copyright holders.