Cacio e pepe is practically synonymous with Pecorino Romano -- but Pecorino clumps when cooked. Add it at the end, instead, which also makes the most of its sharp flavor. Lemon adds spunk -- that is, acidity and freshness.
Use small, slender Persian cucumbers -- they are less watery and have much smaller seeds. Korean hot-pepper powder is a blend of roughly ground sweet and slightly spicy red pepper. Chile de arbol is a good substitute.