The cytoskeleton is a model active matter system that controls processes as diverse as cell motility and mechanosensing. While both active actomyosin dynamics and actin-microtubule interactions are key to the cytoskeleton's versatility and adaptability, an understanding of their interplay is lacking. Here, we couple microscale experiments with mechanistic modeling to elucidate how connectivity, rigidity, and force-generation affect emergent material properties in composite networks of actin, tubulin, and myosin. We use multi-spectral imaging, time-resolved differential dynamic microscopy and spatial image autocorrelation to show that ballistic contraction occurs in composites with sufficient flexibility and motor density, but that a critical fraction of microtubules is necessary to sustain controlled dynamics. The active double-network models we develop, which recapitulate our experimental findings, reveal that while percolated actomyosin networks are essential for contraction, only composites with comparable actin and microtubule densities can simultaneously resist mechanical stresses while supporting substantial restructuring. The comprehensive phase map we present not only provides important insight into the different routes the cytoskeleton can use to alter its dynamics and structure, but also serves as a much-needed blueprint for designing cytoskeleton-inspired materials that couple tunability with resilience and adaptability for diverse applications ranging from wound healing to soft robotics.