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In the past I canceled my Facebook account once, because Facebook is out of control with my private info.
I gave it another try, and I see that I get messages 'soandso has accepted your invitation to be a friend'. I never initiated those invites. How can you control that? It bugs me to the point I want to cancel my account again.

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Seems as though your account has been compromised. There's truly no other reason for it, if you're claiming that you've never sent anyone an invitation. Unless you simply don't remember. This has never happened to me before, and I've never seen any related posts. So the chances of it being a common error are highly unlikely.

The best thing to do, would be to change your registered email and password. Preferably to a newly created email. That way it'd be at much less of a risk.

Account > Settings

Click on "Change" next to Email. Once done, it'll list your existing email and opt you the chance to change it to a new one. You can also edit it to where nobody can see your email. Considering it's displayed at the bottom of the "Info" section of your Facebook. I strongly suggest that you hide it.

You'll now do the same with the password. Type in your old password, your new desired password and then confirm your new password.

This should help. If you're no longer receiving random friends after following the directions above, than someone did have access to your account.

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@Nick You are the man