I used to lay newspaper on the mousetrap, more for my cat than anything, but it also worked on my dogs. Snapping closed with a paper on top does
add another element to the sound, but I think it was more in place so the the trap would be set off without directly touching the trap. I think this because it was dang hard to lay a piece of newspaper on the trap without setting it off myself, so at one point, when I used the traps strictly for dog training and "quarrantined" the cat during these sessions, I stopped using the paper.
Does it hurt the dog? Probably. It hurts me when I accidently set the trap off on myself, but I don't lose the use of my finger or anything and the pain quickly goes away. I would not use it on a frail, small dog, but I haven't had any ill effects on Ridgebacks, which isn't exactly what I would call a dog of bulk, or my Presa, who is light in the bone for her breed.
The thing I like about this method is it only takes a few times, then you don't even need to set the trap anymore. Just placing it where you don't want the dog will go or "nose" will get you the desired response. My pup was getting a little too interested on what was on the coffee table; remotes, magazines..... A snap on the nose (I assume) from a mousetrap and now he looks at the articles, sees the trap setting there, thinks, then finds something else to do. You can almost see the smoke from his mind churning
I am a bit of a believer in what a trainer once told me;
One good correction is worth a thousand nagging ones.
Edited to add: They only sell one size mousetrap around my area. Then you graduate to rat traps. I use the common, wooden mousetrap.