Requirements to write Comments on Posts

What setting did you change to only allow registered users to comment? I’m not aware of any such setting on a per-post basis. I think you just go to Options->General and turn on “Users must be registered and logged in to comment”. It’s global, not per-post.

As for spam, none of these options fight spam. Not really. Spammers don’t use the comment system, as a general rule. What you need is a spam fighting plugin to fight spam effectively. Bad Behavior is very good, as is Akismet. Both in combination stops virtually all spam.

That’s probably a trackback, not a comment. Spammers (ab)use the trackback mechanism to automate submission of their spam. Most themes and such display both comments and trackbacks/pingbacks all in the same space.

Bad Behavior and Akismet both will recognize spam in trackbacks as well as comments.

Right, now that is indeed a per-post setting. I’ll tell you what to change in the database, but I strongly urge you to reconsider this approach and to use spam blocking plugins like Bad Behavior and Akismet instead. Bad Behavior has the nice side effect of blocking spammers from hitting your site at all, thus reducing CPU and database load, while Akismet helps everybody fight spam by communicating with the centralized Akismet server, improving spam recognition methods. With these methods, you can indeed go back to letting anybody post without worrying about spam to any great degree. They work incredibly well.

Still, here’s what you change in the database to disable trackbacks/pingbacks on a per post basis: The wp_posts table should have a column called “ping_status”. Change it to “closed” for all the rows.
Or, in SQL:
UPDATE wp_posts SET ping_status = “closed” WHERE 1=1;