"Quick Links" shows a spider "moderating"
 

Sometimes when I click on "Quick Links" it shows one of the spiders as "moderating" under "Location."

Does that mean they've logged in as a moderator somehow?

I think you mean you selected the page Who's Online from Quick Links...

But anyway, there should be a big red Stop sign next to the spider trying to moderate. That just means the spider tried to go to the page and got told he had no permission.

I didn't see a stop sign. That's what worries me.

Logout and check your permissions then. ;)