Do original Fender parts fit on a Squier?

I know they should, I mean, the Fender Jazz and Squier Jazz have the same dimensions, right?

I was thinking about ''pimping'' my Squier, installing a new pickguard, maybe that thumb rest thingy, put on better tuning pegs, volume pots etc (I've already put in EMG J set pickups, so i thought I should go for complete customization).
I've had a bad experience with an original Fender Jazz bridge, that actually did fit a Squier but the action was impossible.
I know that other stuff I've mentioned won't effect playability but i wouldn't want to waste my time if they're not a perfect match.

I'm also thinking about installing the Badass II bridge.

So, has anyone had experience with all of the above, especially with the Badass II on a Squier (it's a standard jazz series btw)?

And one final thing: I could change my mind and decide to paint the pickguard. In that case, what should I use in the process to not affect the tone?

Thanks

 

Do they sell pickguards with hot babes on them?

I didn't mean literally paint as in with brushes, sorry, I saw that thread where a dude did that Black Sabbath album cover on his pickguard so I had something similar in mind.

And another thing: what are the pros/cons of bridge and pickup covers, if any?