With birds, they'll sometimes eat it and it swells inside them and can cause intestinal blockage. But while I've heard of this concern with mice, I've never known of a mouse to actually it it.
I've used corncob before, but always as one part of a mixture (usually along with aspen and hay). I put a little corncob in the corners, and then cover it with aspen and hay. The only problem I've ever noticed is that sometimes, some bags seem to mold rather quickly when they get wet. But I always change it immediately (because it smells gross).
Also I was wondering, would anything like newspaper or copy paper be ok to put in the very bottom under the substrate, to soak and prevent wetness from being on the cage bottom all the time?
You can generally use newspaper, but they'll pull it up and chew it. You can also use plain baking soda in the corners (where they go to the bathroom) and if you clean the corners out mid-week and replace the corners with clean bedding, often the mice won't go to the bathroom anywhere else. Their instinct tells them to go to the bathroom in a place as far away from the nest as possible so that predators don't smell them, and in a square cage, often that's one or two corners at the other end of the nest.