New advances in many engineering fields require refined modeling strategies to analyze the static and the dynamic behavior of complex structural components and sub-systems with a reduced computational effort. In this context, the present contribution proposes an innovative approach, based on higher-order theories and a unified formulation, to analyze the statics and vibrations of doubly-curved shell structures with irregular domains, including discontinuities and holes of different shapes. The geometry of the panel is described using the principal coordinates, whereas the fundamental equations are derived from the Hamiltonian principle, and are described in their weak form using higher-order interpolations for the unknowns on a discrete computational grid. The constitutive properties of layers are derived from various analytical homogenization techniques, primarily for composite materials, lattice honeycomb and anisogrid cells, Carbon Nanotubes (CNTs), and Functionally Graded Materials (FGMs), among others. Various loading and boundary conditions are modelled, while various methodologies are provided to recover stress and strain components, based on either a Generalized Differential Quadrature (GDQ) and Generalized Integral Quadrature (GIQ) method. Various numerical examples are presented which validate the accuracy and efficiency of the model against the reference solutions from commercial softwares. In addition, the effects of the polynomial-based interpolations of different orders are explored followed by a parametric study aimed at evaluating the sensitivity of the governing parameters to the overall response of structures with different curvatures and materials.