Harry tells Neville to kill the snake, but does not tell him it needs to be done a certain way, or Nagini would not die. Why?
The sword was not needed to kill Nagini. We know this is the case because Harry himself as a horcrux "lite" is not invincible.
All horcrux Items that we encountered were indestructible until a strong enough weapon or force was applied.
Examples of this first rule, can be shown by the locket, it cant be scratched or damaged in anyway.
This Indestructibility would have been either applied fully, or not at all to life form horcruxes and since we have multiple examples of harry himself being cut, burned, breaking limbs, stabbed, etc we can assume that Nagini is also vulnerable to this as well.
However since Harry's not a true horcrux, in the sense that the ritual to create a horcrux was not performed, Nagini herself may have additional protection.
But Voldemort was extremely worried about her welfare during the last battle, so he kept her close and inside a magical barrier. A magical barrier against attacks that could destroy her, however, if she were to be as indestructible as say the locket, this protective spell would not have stopped someone with access to the magic used in destroying a horcrux anyway.
1) He assumed Neville would relay this weird bit of advice to Ron and Hermione, who'd be able to make sense of it.
2) To destroy a Horcrux the vessel must be destroyed beyond magical repair. In the case of inanimate objects this requires extreme measures, like Basilisk venom. However, I'm not sure if it's ever specified how a living host works - a person killed by normal means is still dead beyond magical repair. Perhaps Harry interpreted this to mean that the Nagini-Horcrux would be destroyed when Nagini was killed, regardless of how it was done?
It just occurred to me that when Harry was watching Snape and Voldemort just before Snape died, he tried to think of a curse he could use on Nagini, but knew he would be caught if he missed. So he must have assumed he could potentially kill Nagini without a horcrux. Thus he would not have felt the need to tell Neville to use the sword.