I used mostly cardboard for my worm inn just because it tends not to wad up so much (clumpy). Most certainly for the bottom 'plug'. Think I ended up using up two boxes lol.
For starting any bag I think the sides will always get dry to some extent, and your first harvest might pull some of this unprocessed stuff out but that's no problem (toss back on top). But I've found once you've got lots of VC around, it presses against the side bedding and keeps it moist and then the worms get at it.

So I think cardboard on the bottom and paper on the top should work fine. Avoids the paper getting crushed down into a solid mass if you use it on the bottom.

For how wet, it changes depending on the age of the bag. Certainly much trickier on new bags. When it's 95% bedding I would sometimes poke down a bit to make sure it's still moist in the middle. With that much airflow it can dry out quickly. Once it's going nicely and you've got some solid amounts of VC it's a lot less vital as the VC itself holds so much. Then I just make sure the top is nicely damp with some dribbles of water or use a sprayer. With an older VC loaded system, even light overwatering on top can have it slowly work it's way down through the VC until it's dripping from the bottom. Once you get that it can drip for weeks even if you stop watering.

One thing with heavy watering, it can lead to compression of your VC and bedding. Something like the worm inn is heavy as it is, waterlog it and it could do close to 100lbs. That will compress your VC at the bottom into a mass and when you harvest it it will be like raking clay in your back yard as opposed to it nicely dropping out, not fun.