Could Voldemort have used an elf to Side-along Apparate Harry with him, and take Harry to Voldemort? Since any elf could apparate directly in and out of Harry's house, as seen in Chamber of Secrets.

That is something Voldemort could do, but probably something he would not do. As Dumbledore mentioned in Half-Blood Prince, there are some things (power of love for instance), that Voldemort considers to be below him to be worth giving any attention. As Dumbledore said to Harry In Deathly Hallows (Thanks @Rand al'Thor♦)

"And his knowledge remained woefully incomplete, Harry! That which
Voldemort does not value, he takes no trouble to comprehend. Of
house-elves and children’s tales, of love, loyalty and innocence,
Voldemort knows and understands nothing. Nothing. That they all have a
power beyond his own, a power beyond the reach of any magic, is a
truth he has never grasped."

From DH we also see that Voldemort made a very critical mistake in not realizing Kreacher could have the ability to apparate out of his cave after drinking the potion. Also from Half-Blood Prince, we know that Voldemort likes grandeur in everything he does, which is the reason he wants the Hogwarts founders' personal items for his horcruxes, and not mundane everyday objects. It's also the reason he hatched the elaborate year long plan in Goblet of Fire to smuggle Harry out of Hogwarts instead of trying a simpler more practical approach. So keeping these things in mind,

1. It's possible it never crossed his mind to use a house-elf to apparate Harry out of the Dursley's house (I'm guessing this is where you mean by 'Harry's house'), or it's also possible Voldemort didn't know this could even be done, as he has been shown to be unaware of the details of house-elf magic.




2. The method is simply too boring for Voldemort's tastes. He would rather take out The Boy Who Lived in a manner that would further emphasize his power over the kid.

However, I think the more likely explanation is that it was simply due to the protection offered to Harry at the Dursleys against Voldemort. The Dursleys' house was not protected against apparating, Voldemort could have as well sent any human Death Eater to the house to side along apparate Harry with him. It seems Harry was protected at the Dursleys' house not just directly against Voldemort, but also against any of his goons that might attack him there on Voldemort's orders, as we don't see Voldemort ordering any of the Death Eaters to attack Harry there either. I think that as far as Harry's protection at Privet Drive is concerned, Voldemort ordering a house elf is not any different from him ordering a Death Eater to kidnap Harry, and so the protection magic would kick in to save Harry from anything Voldemort could do

He likely didn’t know a house-elf could.

The Dark Lord didn’t seem to be very knowledgeable about house-elves’ magic. He didn’t realize that Kreacher would be able to Apparate out of the cave where he kept his Horcrux, and Hermione suspects that’s because he wouldn’t think house-elves would have magic he didn’t.

“But … you couldn’t Apparate in and out of that cave,’ said Harry, ‘otherwise Dumbledore –’

‘Elf magic isn’t like wizard’s magic, is it?’ said Ron. ‘I mean, they can Apparate and Disapparate in and out of Hogwarts when we can’t.’

There was silence as Harry digested this. How could Voldemort have made such a mistake? But even as he thought this, Hermione spoke, and her voice was icy.

‘Of course, Voldemort would have considered the ways of house-elves far beneath his notice, just like all the pure-bloods who treat them like animals … it would never have occurred to him that they might have magic that he didn’t.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 10 (Kreacher’s Tale)}}

Dumbledore, who knows the Dark Lord best and was able to figure out all his plans, confirmed he knows nothing of certain types of magic, including house-elf magic, and doesn’t understand their power.

“And his knowledge remained woefully incomplete, Harry! That which Voldemort does not value, he takes no trouble to comprehend. Of house-elves and children’s tales, of love, loyalty and innocence, Voldemort knows and understands nothing. Nothing. That they all have a power beyond his own, a power beyond the reach of any magic, is a truth he has never grasped.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)}}

The Dark Lord didn’t think any type of magic could do something his couldn’t do. So, if he couldn’t Apparate into the Dursleys’ house to get Harry, he wouldn’t think a house-elf could.

No, that wouldn't have worked.

That may be a moot point, since Voldemort did not appear to know that house-elves could apparate into and out of places that wizards couldn't, but even if he had known about that it still wouldn't have worked.

We see at the beginning of Deathly Hallows that Dumbledore's blood charm protects Harry and his household not only from Voldemort but also from his Death Eaters, and even from Stan Shunpike who had no ill intent towards Harry but was under the Imperius Curse. In other words, it protects the residents from anyone working on Voldemort's behalf for any reason. That would certainly include a house-elf acting on Voldemort's orders.

Dobby could apparate directly in and out of Harry's house, despite the blood charm, only because Dobby had no ill intent towards Harry and was not acting on the behalf of anyone who did.

See also this answer for more information about Dumbledore's blood charm, and in particular how it relates to Lily's sacrifice.

Presumably it has something to do with Harry living with actual family. That loving bond the Dursleys evince every day for our young hero!

Sure, the Dursleys did not like Harry, and Aunt Petunia, I believe, despised him in a very special and visceral way that borders on pathological anti-thaumism, the irrational hatred of magical folk. Even so, Harry was safe while living in the Dursley's home. He kept going back every summer holiday. They did provide for him (basic physiological needs, apparently apart from love). They may have hated him, but their presence was a protective magic probably deeper than any mere wand or elf magic could contend with.

We could go one more and ask why not just let Harry grow up in Dumbledore's office? He'd be safe and among wizard folk. But we'd be back to the family connexion issue.

It would be a plot hole - if Voldemort thinks of using house-elf apparition then he would have protected against house-elves being able to access his Slytherin locket Horcrux. Then Kreacher would not have been able to escape the boobytrapped Horcrux location as Voldemort would have considered the idea and put an anti-house elf apparition jinx on the location and Harry Potter would never have found the location of that Horcrux.

It's one of those things that is obvious in hindsight.