I live in a farm situation, and rats and roaches are part of the ecosystem here. We have a dog, who loves to "chase" rats. She doesn't catch them, but harrasses them emough they don't stay around long.
As for roaches in the cardboard stack... We have come to an agreement. As long as the roaches stay outside and do their thing, and don't come inside, they can live there. Roaches are a big part of the decomposer food chain. They break down larger pieces of carbon material, and also feed amphibians, small reptiles and even birds. If you have chickens, bring a few over when you are moving cardboard around, and let them have some roach snacks when you flip the cardboards open.