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?
The move will be Normal-type. Judgement is a Normal-type move, no matter what Pokémon it is used on. Giving a Pokémon any Plate would change Judgment's type according to the Plate.
As an example, In Gen 6 Pure Hackmons Mega Gengar runs Judgement / Secret Sword / Shell Smash / Dark Void, along with a Spooky Plate. This effectively gives the Gengar a 120 BP STAB Judgment, as its typing has now changed to Ghost- because of the Plate. It also makes the Gengar-Mega imposter-proofed, until the Imposter is also carrying any Plate. (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 its Dynamax move was also Bug-type. Now, this also gives us something about Judgment, as it was Normal-type without a Plate, and so was its Dynamax move.
What happens if you Trick or Treat/Forest's Curse a Silvally holding 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.