Is this a common Facebook thing?

I’ve been getting a lot of notifications that people have accepted friend invites that I never sent out. Some are former coworkers, some foaf, some complete strangers. Some are people I like well enough, I just never friended them- I had very few friends, and actually knew all of them. It feels awkward for some reason.

Check to see if someone has spoofed your page and sent out a shitload of invites. It happens to friends of mine all the time.

But the OP wouldn’t get those notifications; they’d go to the spoof account. How are the notifications delivered? I would be highly suspicious of those.

Someone has probably hacked your password and is harvesting data. Change your password immediately, and also change your email password.

Recently, for some reason Facebook has been giving me lots of “friend suggestions” of people I’ve never heard of. I don’t know whether this has been happening to other people or whether your issue is in any way related.

No, it shouldn’t be related. All those recommendations do is allow people to send friend requests, which the OP didn’t appear to do. It’s possible that he clicked on those by accident and sent out a friend request without knowing it, but unlikely.