The national electric code is very specific about these issues, and you would do well to do some research before you start installation. The code book is available at electrical supply houses like Graybar Electric and shouldn't cost a fortune, though you should expect to pay $30 or so. You really shouldn't be getting your supplies from a big box store anyway, as like the paint, the quality isn't necessarily what you can get a Graybar either. Besides, Graybar is more fun, like a toysRus for grownups.
To my off the head knowledge, (and I didn't look this up, so don't go just on this) you need to protect the cable in the wall, usually with metal conduit. Yes, the boxes should be metal. Yes, they need to be screwed into the studs.
Why, you are asking yourself, did I go with metal studs?
At the very least, grommits in the stud openings are a good idea, but like I said, your best bet is to reference the code book.