He can certainly chain Dust Tornado or Mystical Space Typhoon to your activation of Dark World Lightning, but a couple of things will affect the outcome. However, in each case, you still resolve the chain backwards from the last card played.
Dark World Lightning resolves first. If the card you targeted with Dark World Lightning is still face-down (meaning your opponent didn't chain the targeted card), then Dark World Lightning will destroy it and you'll be able to discard. MST/Dust Tornado will resolve next and destroy Dark World Lightning, but not negate it's effect. Destroy does not equal negate.
The important things to remember about chains is that 1) they resolve backwards starting withethe last card played, 2) all cards in the chain remain face-up on the field until the chain fully resolves and 3) if a card is destroyed/removed from the field mid-chain, that does not remove it's effect from the chain. Only negation can remove an effect from a chain, not destruction.