Pokemon SS Help: Judgment and STAB (read description)?

Yes, you would get STAB from that. One of the secondary effects of Spooky Plate is that it makes judgement a ghost type move, hence you would not only get STAB, but also a secondary power boost due to spooky plate.

Arceus would also get STAB.

Well natures are very important and they actually do matter. There is absoulutely no reason to run all 4 of those fire moves on one typhlosion. Flame thrower or fireblast, swift, sunny day, and solar beam would be a good set. Golems magnitude is completely outclassed by earthquake, so I would take that of and replace it with something like rock polish to give him a chance to sweep through entire teams or explosion as a last resort to deal huge damage. There is no reason to have blizzard and Ice fang on gyrados. He is much better with physical attacks so take of blizzard and replace it with something like stone edge or rock slide. Your team has a large water weakness, so an electric type should help balance your team out(you could add a grass but it would give you a big ice weakness) So an ampharos(evolve from mareep) or a lanturn(from chinchou) would both compliment your team well.

For your first question, I'm guessing it will since the info for judgment is it changes type according to the held plate.

and for your second question, arceus ALWAYS gets STAB for judgment because it always changes it's type to the plate's and judgment also changes it's type into the plate. If you had no plate, judgment would be normal and arceus would be normal so it will still get STAB.

Jugdement would as its type is plate dependent.