FBI never said wood stock, they said it was a Mauser in 30-06.
The also never claimed it was a take down, there’s only two screws holding the stock to the action and they can be broken down and re assembled in less than a minute.
Bullet didn’t hit a vest, it was most likely something light like a 125gr or 150gr bullet at high velocity that fragmented on impact and the pieces followed the spines. I’ve had it happen with deer and there is historical precedent as MLK had the same thing happen when he was killed with a 30-06 out of a Remington 760.
As for how they know it was that rifle, it was found stashed in a small wooded area with a fired round in the camber less than 300 yards from where the assassin jumped off the roof, in the direction that he ran with the rifle as seen on campus security camera and the direction he was seen coming from on the local ring camera without the rifle in his possession. Additionally if I understand correctly it was found by the dogs who were following the shooters path from where he jumped off the roof.