So I was reading the GF Legacy book, and it explains that Luca Brasi was the one who snuck into Woltz's home with the horsehead. How did he manage that...thoughts?

M-O-N-E-Y.

Where does it say in "The Godfather Legacy" that Luca killed the horse?

Yeah, where?

Also, Luca was kept pretty busy. Not only was he directed to find out about Sollozzo, but he had to fly out to California to decapitate a horse. Madonne!

They explain it on page 34.

If I'm not mistaken it's implied that Luca had something to do with it, but nothing indicates that he did it himself; he could have just got some cronies that were completely reliable (or, more likely, just scared shitless of what Luca would do to them if they told anyone). I don't have the novel on hand to check but IIRC it's something just like the Don asked that Luca Brasi see him, and that Hagen was confident that afterwards Woltz would inform them that Johnny had the movie part. The only part that could possibly indicate Luca did it himself is the note that whoever hacked off Khartoum's head must have been very strong, and perhaps the fact that it was hacked off with an axe (since in the flashback portion of the novel, Luca shows himself to be very handy with an axe as one unfortunate Capone henchman could attest, and the novel IIRC mentions that Luca was pretty strong)

While it's not canon by any means, the video game has your character (Aldo) and Luca go to the mansion; Luca goes into the stable and kills the horse and then you deliver it to Woltz's bedroom. Again though that's obviously not canon.

I don't know where this is coming from about Luca and the horse. The novel says nothing about who did it. Woltz assumed that Vito had ordered it and he called Tom and threatened Tom afterward. That's all.

As it turns out:

In a deleted scene just before Vito, Tom and Sonny discuss the upcoming Solozzo meeting, Vito asks Tom about his meeting with Woltz. After Tom tells Vito that Woltz is "not a Sicilian," Vito says, "Send Luca Brazi to me. I think we're going to find a way to reason with this Mr. Jack Woltz."

It's true that the novel doesn't identify Luca with the horse-killing. And, the deleted scene doesn't prove that Luca actually killed the horse--perhaps he organized it, but didn't wield the fatal axe. But, I can't blame Harlan Lebo for concluding that Luca did it.

he most likely paid off the guards

This is one of those unfortunate mistakes FFC may have made. Luca was really no good at organizing anything. He couldn't even bluff Tattaglia Junior when he tried. Seems to me this was a job for Tessio or Clemenza. As for Luca...he could have ripped the horse's head off!

One thing that could be relevant: in the novel, Michael tells Kay that Luca is known for the ability to plan a murder solo - no help, no witnesses.