What happens with continous trap/spell cards when they get negated by Light and Darkness Dragon?
These rulings come from yugioh.wikia.com, and note:

These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami.[18] They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise. References: [19][20][21][22][23][24][25][26]

The relevant rules are:

“Light and Darkness Dragon” cannot negate the activation of the Ignition-like effect of Continuous Spell or Trap Cards (“Ultimate Offering”, etc.) or Field Spell Cards (“Ancient City - Rainbow Ruins”, etc.). “Light and Darkness Dragon's” second effect activates when a Spell or Trap Card is first activated (placed face-up onto the field or flipped face-up).17
When “Light and Darkness Dragon” negates the activation of a Continuous Spell or Trap Card, the Continuous Spell or Trap Card is sent to the Graveyard. When “Light and Darkness Dragon” negates the activation of “Swords of Revealing Light”, the “Swords of Revealing Light” is sent to the Graveyard.17