Electrical wires typically run either vertically, up and down the side of a stud (with staples), in order to reach receptacles, ceiling lights/fans, etc., and horizontally in order to get across the room(s). The vertical wires are typically pretty easy to avoid: avoid drilling/nailing above a receptacle or light switch, or, if you have to, avoid missing on the side that the receptacle is nailed to. The horizontal runs should have enough play in them to avoid most damage, unless you drill/nail into the hole in the stud (called a nipple) that they pass through.
I don't think that it's required that you put any metal plate on the stud in order to protect the Romex/cable/conduit, but rather that it is only required if you drill the nipple too close to one side of the stud, at which point a metal brace is needed in order to ensure structural strength. Outside of drilling/nailing into an unprotected nipple, or very near it, there is little to worry about when it comes to the electrical.
When it comes to pipe, you should be able to tell if you hit copper pipe. Even though it might be one of the softer metals, it's still going to offer a substantial amount of resistance, and unless you hit it where it passes through a stud, your nail/drillbit will probably deflect off of the curved surface of the copper pipe. With PVC or ABS, however, yeah, you're most likely going to have a leak if you hit it squarely with a drillbit, maybe even a nail.
When it comes to cutting large holes in drywall, cut horizontally first -- if there's a stud or vertical pipe, it's better for you to find it immediately, at which point you might decide it's better to make a new hole on the other side of the stud, rather than later, after you've already made a long vertical cut in the drywall.