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On negation,
It will only activate once per chain otherwise it negates itself (uneccessary to explain now because of PSCT)
It is mandatory, so it will always try to respond to the first effect it can (either a spell speed 1/2 effect activated on chain link 1 outside of segoc chains or the last effect of a segoc chain, assuming in both cases it is not an effect like super polymerization that cannot be chained to). Even if multiple mandatory effects try to do that (Ladds, doomcals) they will all activate and attempt to negate the same effect.
When an effect tries to negate "an effect" directly (not negate "the effects" like veiler etc. I mean ladd, shi en, you know what I mean) it can only negate if it's directly after it on the chain. That means that if an effect activates and multiple ladds try to negate it, they will all be put on the chain sequentially by segoc rules, all attempting to negate that first effect, but only the one directly after it will work, negating, and the rest will fizzle and not lose any atk.
LaDD can negate the initial activation of continuous spell/trap cards, but not the effect of an already face up spell or trap card. If such an effect activates, ladd will not negate.
If the last link of a segoc chain is the effect of a face up spell/trap card, for example you summon summoner monk while your opponent has black garden face up, ladd will add itself to that chain, chaining to the last effect on the chain that can trigger it. In the above example, since it does not care about Black Garden, it will attempt to negate the summoner monk that is trying to switch to defense, forming the chain CL1: monk, CL2: garden, CL3: Ladd. However as it was mentioned you can only negate something that is directly before the negating effect in the chain, so since the irrelevant black garden is the effect before ladd, ladd will resolve without effect, unable to do anything.
Since it's mandatory, it will activate even if it cannot reduce ATK/DEF further, however it needs to be able to reduce BOTH 500 ATK and DEF at resolution in order to negate (if it didn't, it would always be able to negate), otherwise, if it is not able to reduce both 500 atk and def for any reason (either stat is below 500, or LaDD is no longer face-up on the field) it will just resolve without effect. Currently how this works is explained by the "and" conjunction in its new text. As psct stands, "and" means that you do both at the same time, and you have to be able to do both to be able to do either. http://www.konami.com/yugioh/articles/?p=4514
It negates but does not destroy. If it negates Battle Fader, it stays in the hand and can be activated during the next attack declaration. If it negates the effect of Formula Synchron to synchro summon, Formula Synchron will remain on the field, and if the materials are still there Formula Synchron can activate again, until ladd can no longer negate it. Lightsworn monsters that mill in the end phase will keep activating until they resolve the effect to mill, ladd will keep negating until it is no longer able to negate.
However, continuous cards (continuous spells/traps, equip spells, field spells) need to resolve properly initially to gain the properties that allow them to remain face up, so if the initial activation of a continuous card is negated, that card will be sent to the graveyard by game mechanics.

On destroying and special summoning:
The effect to destroy and special summon is mandatory. It will always activate.
Even if you can't do one of the two (destroy or special summon) you will still do the other, additionally both are considered to happen at the same time for timing purposes. That is indicated in the psct text by the "also" conjunction, how it works is explained in the above article.
So long as it is destroyed, anywhere, the effect will activate. That means even if it is destroyed while in the hand (fire king avatar yaksha, good ol' crush card virus), in the process of being summoned (solemn warning), or while in the s/t zone (equipped by relinquished and destroyed when relinquished's effects get negated). Remember though that cards in the graveyard, banished cards and cards in the deck cannot be destroyed (don't say chain destruction, it's an exception because the card text specifically states so. for the record, it triggers in that case as confirmed by the old sacred phoenix ruling).

On double attribute:
It is treated as both LIGHT and DARK at the same time. You can use Honest with it (will be negated if ladd can still negate), you can tribute it for a virus if the attack when you activate the effect is the required one (since ladd is tributed as a cost, it will not negate since it is not face up at resolution to reduce atk/def). When gozen match is on the field, ladd is destroyed if its effect which makes it dark is active.