To answer the question in your heading - visiting babies, geese, turkeys, peacocks and occasionally even dogs :D
If a puppy needs to eliminate it will, regardless of where it is or whether paper is showing. I currently have an empty crate in with my puppies as if one is a bit slow eating I will shut him or her in there to finish their meal in peace - the bitches especially will also go in there to wee

On a more serious note I just put bedding in the crates, if you use fleece/vetbed and your pup does have an "accident" then it will soak though and the top remains dry. If you put water in, make it secure - easier said than done - as pups invariably tip it over.
If the crate is much larger than your pup and you only intend him to sleep in there you can temporarily make is smaller by putting a cardboard box or two in there to take up some of the space.
HTH