Nanoparticles (NP) functionalized with single-stranded DNA (ssDNA) offer a route to custom-designed, self-assembled nanomaterials with potentially unusual properties. The bonding selectivity of DNA guarantees one-to-one binding to form double-stranded DNA (dsDNA), and an appropriate base sequence results in head-to-tail binding linking NP into networks. We explore the phase behavior and structure of a model for NP functionalized with between 3 and 6 short ssDNA through simulations of a coarse-grained molecular model, allowing us to examine both the role of the number of attached strands (valency) and their relative orientations. The NP assemble into networks where the number of NP links is controlled by the number of attached strands. The large length scale of the DNA links relative to the core NP size opens the possibility for the formation of interpenetrating networks that give rise to multiple thermodynamically distinct states. We find that the 3-functionalized NP have only a single phase transition between a dilute solution of NPs and an assembled network state. 4-Functionalized NP (with tetrahedral symmetry) exhibit four amorphous phases, or polyamorphism, each higher density phase consisting of an additional interpenetrating network. The two investigated geometries of 5-functionalized NP both exhibit two phase transitions and three amorphous phases. Like the 4-functionalized NP, the highest density phase consists of interpenetrating networks, demonstrating that regular symmetry is not a prerequisite for interpenetration to produce thermodynamically distinct phases. The width of the coexistence regions for all phase transitions increases with increasing functionality. Finally, for 6-functionalized NP with octahedral symmetry, the possibility of observing disordered phases with significantly bonded particles is preempted by the formation of ordered crystal phases. Interestingly, the extreme softness of the potential combined with the directional interaction allows for the formation of (at least) six distinct crystalline structures (i.e., polymorphism) consisting of up to six interpenetrating simple cubic lattices.