The short answer... We address the heap one criterion at a time.
Each piece of criteria we've been given is a problem to solve. One problem with many possible solutions. So, step one is to solve for each design requirement.
Step two is roughing out how all of the individual criteria will come together to form one cohesive product concept. The key is rough. 10+ criteria are a lot to juggle mentally and visually, so utilizing a rough fidelity allows us to form a cohesive concept with the least amount of friction.
Step three is to polish up the design. (figuratively) Now that we've got all the solutions blocked out to fit together as a whole, we go in and refine the overall design.
How do you design amidst a heap of product criteria?