If i have a izzet staticaster paired with nightshade peddler . As AP i “ping” a 5/5 with the staticaster and in response my opponent kills the staticaster. Will that damage done to the 5/5 be deathtouchy?

Yes it will. Since the Staticaster is not on the battlefield when the ability goes to resolve, we use the last-known information about it. When it was last on the battlefield, it had deathtouch, so the damage the ability would deal is coming from a source with deathtouch. So the damaged creature(s) will be destroyed.

Nathan Long
Wizards.com Boards NetRep

608.2g is the principle rule that makes it so
there's also 702.2e as a “reminder” rule to reinforce the interaction

608.2g If an effect requires information from the game (such as the number of creatures on the battlefield), the answer is determined only once, when the effect is applied. If the effect requires information from a specific object, including the source of the ability itself or a target that’s become illegal, the effect uses the current information of that object if it’s in the public zone it was expected to be in; if it’s no longer in that zone, or if the effect has moved it from a public zone to a hidden zone, the effect uses the object’s last known information. See rule 112.7a. If an ability states that an object does something, it’s the object as it exists—or as it most recently existed—that does it, not the ability.

702.2e If an object changes zones before an effect causes it to deal damage, its last known information is used to determine whether it had deathtouch.

And if the Peddler is destroyed while the ability is on the stack, the damage won't kill the 5/5, right?

Jep.

These interactions were covered in the Rules Tip Blog about a month ago. Although aimed mainly at players, the blog is also useful for judges. :)

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