Does "Thunder Dragon's" Effect Activate If Discarded By Cards Such As Raigeki Break, Lightning Vortex, or Delinquent Duo?

No. The effect must be activated by the controller to get the effect. Discards don't count.

The effect pretty much tells you.

Thunder Dragon

Discard this card from your hand to the Graveyard to add up to 2 "Thunder Dragon" cards from your Deck to your hand. Your Deck is then shuffled. This effect is activated only during a main phase.

As you notice it says "Discard this card from your hand to the Graveyard" it does not say "If this card is discarded from your hand to the Graveyard"

Happy Dueling

Yeah, you can't "mix" effects like that unless forced to like if Makyura is discarded from your hand with Lightning Vortex or something or states otherwise.