NSA follows ASA in regard to thrown bats. Not all thrown bats are treated equally!
You can have a carlessly discarded bat where the batter accidently let's go of her bat after a swing or tosses it with too much force. NSA and ASA provide no penalty for a carelessly discarded bat accidently thrown too hard or too far. Live ball, play on.
If any discarded bat interferes with a play, then it is interence, the ball is dead and the batter is out.
If a bat is purposely thrown, as in anger, then the unsportsmanlike conduct penalty can be applied- same as with any other thrown piece of equipment. Unsportsmanlike conduct may be punished by ejection- at the umpire's discretion- but no out is charged. The umpire may issue a warning if the infraction is judged minor.
For the situation in the first post, it sounds like the carelessly discarded bat rule would apply. An umpire could issue warnings just as a safety measure and maybe talk to the batter's coach to see if he could speak to the player. And that's about it.
The only other option would be to somehow judge the act as intentional, in which case he could eject the batter. Calling this intentional might be a stretch. That would kind of be a "last resort" ruling if this was judged dangerous enough. But it's not really what the rule is designed to address. Unsportsmanlike bat throwing is usually something like the batter heaving the because she was mad, struck out or made an out.