If you hacked/found a way to get Judgment on, say, a Tyranitar. What typing would it be? Would giving Tyranitar a Plate change the typing of Judgment?
Yep; it'd be Normal- type. This is because Judgement is a Normal- type move, no matter on what Pokémon it is used, and giving the Pokémon a Plate, any Plate, would change it's typing in correspondence with the Plate.
As I'm not citing any source here except for my word, I'll give an example.
In Gen 6 Pure Hackmons, if you've ever played that, you'd have had seen Gengar-Mega with Judgement / Secret Sword / Shell Smash / Dark Void, in conjunction with a Spooky Plate. Now this gives the Gengar-Mega a 120 BP STAB Judgment, as it's typing has now changed to Ghost- because of the Plate, and also it makes the Gengar-Mega imposter-proofed, untill and unless the Imposter is also carrying any Plate, as a non-Plate Judgment is always Normal- type.
What happens if you give Multi Attack to a random Pokemon holding a memory?
Check this replay. Over here, the Multi Attack was Bug- type, and it's Dynamax'd move was also Bug- type. Now this also gives us something about Judgment, as it was Normal- type without a Plate, and so was it's Dynamax move.
What happens if you Trick or Treat/Forest's Curse a Silvally holding a memory and use MA?
Check this replay. Over here, Trick-Or-Treat was used on a Silvally-Dark, and it still hit the Gourgeist super-effectively, and even after Forest's Curse the Phantump was hit super-effectively. So, the typing of Multi Attack won't change and would remain the same.
Hope this helps.