It counts against the dropper, not the dropped-upon. So if a player leaves the game, it's not your fault the thread was dropped so it doesn't count against you.
It's against whoever dropped the tag, regardless of which party started the thread. Like, if you tag into someone else's thread, and they never tag back, they dropped the thread and you're welcome to put the dropped tag on it at the point you consider it "too long" (after a week), even if we can't do anything about the header.