Minor Conjuration does not duplicate objects
The relevant text states, (PHB 116, emphasis added):
Starting at 2nd level when you select this school, you can use your action to conjure up an inanimate object in your hand or on the ground in an unoccupied space that you can see within 10 feet of you.
This text does not say that you can conjure an exact copy of something; only that you can conjure something at all.
In the case of your sheet of paper, you can conjure a sheet of paper that looks exactly like the unknown one, but you wouldn't be able to verify that the writing on the other side actually matches the unknown's. Likewise, for the small chest, you will summon a chest with a latch, but it may or may not be the same as the target chest.
Given that there isn't much detail in the text of the rules, it will be up to the DM to decide exactly how accurate that conjuration will be. For example, I would rule that the object conforms to the caster's desires: if the caster wants to see a piece of paper with incriminating evidence, they will conjure one, even if it differs from the real sheet of paper. Such a ruling is consistent with the fact that the conjurer has control over what the conjured object is.
It's also worth adding that minor conjuration is worthless for forgeries, because it requires the conjured object to be nonmagical, but
The object is visibly magical, radiating dim light out to 5 feet.