Yes
All perfectly normal! When you say bunched up paper stuff what is it exactly? Carefresh? The best material for nesting (not the same as bedding/substrate) is plain white toilet paper torn into strips. They really like that for nesting and it's safe if accidentally swallowed. Also cheap and easy! So individual sheets of toilet paper torn into strips and a big pile of it in the cage (not in the house because as you've seen they do like to forage for it to take to the nest - they pouch whatever they forage for to take to the nest).
They will also hoard under their nest so be careful not to overclean as this stresses them and leads to abnormal behaviours like peeing on their hoard (to deter people from stealing it). If the nest and hoard haven't been pee'd on they can be left for quite a long time really - I never remove the nest if it's dry. The rest of the cage you can "spot clean" mostly then you only need to change the substrate every 6 to 8 weeks or even longer.
So partial cleaning is best - don't clean everything at the same time - eg wheel one week, any toys etc a different week, any substrate a different week again. And replace a bit of the old substrate on top of the new so it still smells familiar. That way something always smells familiar to them and it doesn't stress them.