In Essence Divided, meaning essentially...what?

If Harry has a connection with Horcruxes, then why could he not sense the fake one in the cave, or use a type of "spidey sense" to figure out where the others are- or recognize the diary.

However, that could all be nullified if Harry is only connected to living Horcruxes.

He might be aware of them to an extent without realizing it. He admitted to being overly obsessed with the diary without knowing why (he kept rifling through it despite knowing there was nothing in there, and he wouldn't throw it out despite Ron telling him he should chuck it). I read the quote recently, but don't have the book on me at the moment. It didn't appear like an overly obvious thing like with Nagini, which might be because Voldemort was actually possessing her at the time, I'm not sure. It just didn't seem like it was strong enough for the connection to make itself known to him without him consciously realizing it's there already. But it's an interesting thought that he might be able to use something like that in the future, to find/destroy the remaining horcruxes.

On a separate note, I'm intrigued by how Harry said he felt as if a snake were rising within him the couple of times he came into eye contact with Dumbledore after that. There was the time immediately following the snake's attack, when he was grasping the portkey, but there was also another time later on, I think when he was in Dumbledore's office again after Marietta had snitched about the DA. I just find it interesting that he's feeling Voldemort's emotions toward Dumbledore at those times, but that Harry feels it as if it's a snake's emotions. I wonder what everyone thinks about this - is he feeling Voldemort's true emotions, and they happen to be snakelike when it comes to Dumbledore? Or is it perhaps evidence of Voldemort being a part of Nagini, and he's feeling more of the snake than of the man at those times?

There's a point somewhere when Harry's viewing the stories in the diary where he finds the scene familiar, or finds something about Tom Riddle familiar.

I just scanned the first diary scene when we are introduced to young Riddle, and there is no mention of Harry finding Riddle familiar.

The only time he seems to find a connection to Riddle is when he first gets the diary and cannot put it away, here is the quote:

COS, pg 174, Brit. Ed

I don't know which scene you're talking about Firefly, perhaps if you can find it that would also help.

The scene above is more telling because he hadn't heard of Riddle before now. The only way he would know the name is if when he was really tiny Dumbledore has let it slip in front of an audience who LV really was before his 'coming out party' so to speak.

So why does the name seem to strike a chord with Harry? If Harry does have a stronger connection with LV, perhaps he knows the name because inherently he knows about LV and can get a sense of him. This name means something to Harry because he is essentially connected to LV.

She did say CoS holds a lot of clues, and I think this is definately one.