"Soft and squishy" is a bulk property. That's not due to either the polyurethane or the air fraction individually, but of the combination.
However, at the scale of a knife's edge, that foam is in fact locally either air or polyurethane. Cutting the air bubbles is trivially easy, but that small fraction of polyurethane is actually quite tough. So, while you're putting only a few newtons of force on the knife, at the edge those few newton are concentrated on only a few points. Let's assume that each point where you're cutting the foam is 0.1mm square and the force locally is 0.1 N. That gives a pressure on the edge of 10 MN/m2, or 10.000 bar. Quite a lot.
Paper, especially when freshly cut, might appear to have smooth edges, but in reality, its edges are serrated (i.e. having a jagged edge), making it more like a saw than a smooth blade. This enables the paper to tear through the skin fairly easily. The jagged edges greatly reduce contact area, and causes the pressure applied to be rather high. Thus, the skin can be easily punctured, and as the paper moves in a transverse direction, the jagged edge will tear the skin open.
Paper may bend easily, but it's very resistant to lateral compression (along its surface). Try squeezing a few sheets of paper in a direction parallel to its surface (preferably by placing them flat on a table and attempting to "compress" it laterally), and you will see what I mean. This is analogous to cutting skin with a metal saw versus a rubber one. The paper is more like a metal one in this case. Paper is rather stiff in short lengths, such as a single piece of paper jutting out from a stack (which is what causes cuts a lot of the time). Most of the time, holding a single large piece of paper and pressing it against your skin won't do much more than bend the paper, but holding it such that only a small length is exposed will make it much harder to bend. The normal force from your skin and the downward force form what is known as a torque couple. There is a certain threshold torque before the paper gives way and bends instead. A shorter length of paper will have a shorter lever arm, which greatly increases the tolerance of the misalignment of the two forces. Holding the paper at a longer length away decreases this threshold (i.e. you have to press down much more precisely over the contact point for the paper to not bend). This is also an important factor in determining whether the paper presses into your skin or simply bends.
Paper is made of cellulose short fibers/pulp, which are attached to each other through hydrogen bonding and possibly a finishing layer. When paper is bent or folded, fibers at the folding line separate and detach, making the paper much weaker. Even if we unfold the folded paper, those detached fibers do not re-attach to each other as before, so the folding line remains as a mechanically weak region and decreasing its stiffness. This is why freshly made, unfolded paper is also more likely to cause cuts.
Lastly, whether a piece of paper cuts skin easily, of course depends on its stiffness. This is why office paper is much more likely to cut you than toilet paper. The paper's density (mass per unit area), also known as grammage, has a direct influence on its stiffness.
Best Answer
Sand paper removes material. When used properly, that removal of material can make a blade sharper. However, when cutting the sandpaper, there is no attempt to structure the removal of the material. It will simply dull the scissors. It will remove material in a relatively haphazard manner, taking off the sharp edge.
If you have any questions of this, ask someone who sews for their nice fabric scissors, and let them know you're going to go cut some paper with them. Find out how quickly they respond in an effort to avoid dulling their scissors. Perhaps its not the most scientific approach, but it is a well documented one, and very evidence based! And that's just normal paper!