I work with 8 of them everyday, so they're definitely still used. We're getting a new shovel in January from what we hear, so it'll soon be 9 of them.
Anyhow, like the video Lashlander posted shows, the door is closed by gravity. As the bucket is lowered and drawn back to start another pass, the weight of the door and the momentum of the movement push the door closed and allow the latchbar to lock into the dutchman. The latchbar is a large bar that runs vertically inside the door, and gravity as well as the weight of the latchbar and latchbar lever hold the bar down. The dutchman is a hole in the bottom of the bucket that the latchbar hooks into and hold the door closed. When you want to dump the bucket, you move the lever inside the cab, or push the button on the joystick as is the case on the newer P&H controls, and that engages the trip function. That means that the winch mounted at the base of the boom pulls on the small cable called a trip rope. That rope is attached to the latchbar lever which runs horizontally across the back of the door and pulls up in the latchbar, which then lifts out of the dutchman and the door swings open. To put it in simpler terms to grasp, think of it as being like a door latch on a door in your house. Now, instead of a round door knob, imagine you have a lever type door handle. When you move the handle, the latch pulls into the door and out of the corresponding hole in the door frame...and the door swings open. Very similar to that.
Now, as for the door not hitting the side of the box......well, it happens. Mostly it has to do with the operator. If you are any good, you shouldn't be hitting the truck box at all. That being said, sometimes it happens.....and for some, it happens alot...lol.
There are systems out there like P&H's Snubrite system. It's basically a big fluid powered snubber that helps to slow the door down through resistance and leverage. Keep in mind, the door opens and material starts to fall out before the door is all the way open, so it helps to lessen the impact a bit. Also, you should be tripping the door fairly low above the box......unless you're mad at the trucker, then it's bombs away with some folks.
I've attached a picture a friend sent me from down in the states that shows the back of a bucket. We use a different style bucket, but I have no good pictures that show the back side of them so you can see the different components. Sorry about the labelling I put on it, I could have done a better job.
I'll also attach a picture of one of ours that shows the bottom of the bucket so you can see the dutchman. Also another pic that shows a Snubrite setup. There are two of those on each P&H bucket.
Hopefully that helps to answer the questions you have, but feel free to ask any more and I'll try to do a better job of answering....lol.