I've had my MIA Strat since February and I've only just noticed bits of plastic sticking up from underneath some of the screws in the pickguard. I started yanking it and now I've got a ****load of plastic coming off, but I think I'm supposed to remove it anyway because the pickguard looks fine without the thin plastic on it...

Am I suppose to remove it? (there's some on the back plate as well)
That's just a protective coating so it doesn't get scratched before you buy it. It's ok to take it off. The pick guard will much more shiny without it.
that comes on everything, watches, cell phones
"Spin the middle side topwise. Topwise!"

"And there's Jimmy Page, the biggest thief of American Blues music"
I once saw a new Volkswagen with a blue protective plastic hanging off of its rims after the guy had brought it home haha.

But anyways that is put there to protect the guitar during shipping from scratches, you're supposed to take it off.
Carvin DC145
Epiphone Explorer with Seymour Duncan JB and '59
Ibanez RGT42FXQM with Dimarzio Evo 2 and PAF Joe
Carvin V3 2x12 combo with JJ 6L6GC tubes

Ibanez SR300IPT
Carvin BX600 2x10 half stack
Yank it all off. Also you might have to take some knobs off to be able to get the plastic out from under them. I wish they didn't put the plastic on the pickguards