Recent advances in the study of synthetic dimensions revealed a possibility to employ the frequency space as an additional degree of freedom which allows for investigating and exploiting higher-dimensional phenomena in a priori low-dimensional systems. However, the influence of nonlinear effects on the synthetic frequency dimensions was studied only under significant restrictions. In the present paper, we develop a generalized mean-field model for the optical field envelope inside a single driven-dissipative resonator with quadratic and cubic nonlinearities, whose frequencies are coupled via an electro-optical resonant temporal modulation. The leading-order equation takes the form of a driven Gross-Pitaevskii equation with a cosine potential. We numerically investigate the nonlinear dynamics in such a microring resonator with a synthetic frequency dimension in the regime where parametric frequency conversion occurs. We observe that the modulation brings additional control to the system, enabling one to readily create and manipulate bright and dark dissipative solitons inside the cavity. In the case of anomalous dispersion, we find that the presence of electro-optical mode coupling confines and stabilizes the chaotic modulation instability region. This leads to the appearance of an unconventional type of stable coherent structure which emerges in the synthetic space with restored translational symmetry, in a region of parameters where conventionally only chaotic modulation instability states exist. This structure appears in the center of the synthetic band and, therefore, is referred to as the band soliton. Finally, we extend our results to the case of multiple modulation frequencies with controllable relative phases creating synthetic lattices with nontrivial geometry. We show that an asymmetric synthetic band leads to the coexistence of chaotic and coherent states of the electromagnetic field inside the cavity, i.e., dynamics that can be interpreted as chimeralike states. Recently developed chi((2)) microresonators can open the way to experimentally exploring our findings.